My Favorite Proof of the Cauchy–Schwarz Inequality

December 10, 2024 by Ethan N. Epperly 1 Comment

The Cauchy–Schwarz inequality has many proofs. Here is my favorite, taken from Chapter 3 of The Schur complement and its applications; the book is edited by Fuzhen Zhang, and this chapter was contributed by him as well. Let $x,y \in \real^n$ be vectors, assemble the matrix $B = \onebytwo{x}{y}$ , and form the Gram matrix

$A = B^\top B = \onebytwo{x}{y}^\top \onebytwo{x}{y} = \twobytwo{\norm{x}^2}{y^\top x}{x^\top y}{\norm{y}^2}.$

Since $A$ is a Gram matrix, it is positive semidefinite. Therefore, its determinant is nonnegative:

$\det(A) = \norm{x}^2\norm{y}^2 - |x^\top y|^2 \ge 0.$

Rearrange to obtain the Cauchy–Schwarz inequality

$|x^\top y| \le \norm{x}\norm{y}.$

Equality occurs if and only if $A$ is a rank-one matrix, which occurs if and only if $x$ and $y$ are scalar multiples.

I like this proof because it is perhaps the simplest example of the (block) matrix technique for proving inequalities. Using this technique, one proves inequalities about scalars (or matrices) by embedding them in a clever way into a (larger) matrix. Here is another example of the matrix technique, adapted from the proof of Theorem 12.9 of these lecture notes by Joel Tropp. Jensen’s inequality is a far-reaching and very useful inequality in probability theory. Here is one special case of the inequality.

Proposition (Jensen’s inequality for the inverse): Let $x_1,\ldots,x_n$ be strictly positive numbers. Then the inverse of their average is no bigger than the average of their inverses:
$\left( \frac{1}{n} \sum_{i=1}^n x_i \right)^{-1} \le \frac{1}{n} \sum_{i=1}^n \frac{1}{x_i}.$

To prove this result, embed each $x_i$ into a $2\times 2$ positive semidefinite matrix $\twobytwo{x_i}{1}{1}{1/x_i}$ . Taking the average of all such matrices, we observe that

$A = \twobytwo{\frac{1}{n} \sum_{i=1}^n x_i}{1}{1}{\frac{1}{n} \sum_{i=1}^n \frac{1}{x_i}}$

is positive semidefinite as well. Thus, its determinant is nonnegative:

$\det(A) = \left(\frac{1}{n} \sum_{i=1}^n x_i\right) \left(\frac{1}{n} \sum_{i=1}^n \frac{1}{x_i}\right) - 1 \ge 0.$

Rearrange to obtain

$\left( \frac{1}{n} \sum_{i=1}^n x_i \right)^{-1} \le \frac{1}{n} \sum_{i=1}^n \frac{1}{x_i}.$

Remarkably, we have proven a purely scalar inequality by appeals to matrix theory.

The matrix technique for proving inequalities is very powerful. Check out Chapter 3 of The Schur complement and its applications for many more examples.

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Rejection Sampling

October 8, 2024 by Ethan N. Epperly 3 Comments

I am delighted to share that my paper Embrace rejection: Kernel matrix approximation with randomly pivoted Cholesky has been released on arXiv, joint with Joel Tropp and Rob Webber.

On the occasion of this release, I want to talk about rejection sampling, one of the most powerful ideas in probabilistic computing. My goal is to provide an accessible and general introduction to this technique, with a more “applied math” rather than statistical perspective. I will conclude with some discussion of how we use it in our paper.

Motivating Application: Randomized Kaczmarz

Consider the task of solving a linear system of equations $Ax = b$ , where $A \in \real^{n\times d}$ is a matrix and $b \in \real^n$ is a vector. For simplicity, we assume that this system is consistent (i.e., there exists a solution $x_\star$ such that $Ax_\star = b$ ). The randomized Kaczmarz algorithm is a mathematically elegant algorithm for solving this problem, due to Strohmer and Vershynin. Beginning with initial solution $x \gets 0$ , randomized Kaczmarz repeatedly performs the following steps:

Sample. Generate a random row index $I$ according to the probability distribution $\prob \{I = i\} = \norm{a_i}^2 / \norm{A}_{\rm F}^2$ . Here, and going forward, $a_i^\top$ denotes the $i$ th row of $A$ and $\norm{\cdot}_{\rm F}$ is the Frobenius norm.
Update. Set $x \gets x + (b_I - a_I^\top x)\cdot a_I / \norm{a_I}^2$ .

Convergence theory of the randomized Kaczmarz iteration is very well-studied; see, e.g., section 4.1 of this survey for an introduction.

For this post, we will consider a more basic question: How can we perform the sampling step 1? If $n$ and $d$ are not very large, this is an easy problem: Simply sweep through all the rows $a_i$ and compute their norms $\norm{a_i}^2$ . Then, sampling can be done using any algorithm for sampling from a weighted list of items. But what if $n$ is very large? Suppose, even, that it is computationally expensive to read through the matrix, even once, to compute the row norms. What can we do now?

Here is one solution is using rejection sampling, due to Needell, Ward, and Srebro. Suppose that we know a bound on the row norms of $A$ :

(1) $\norm{a_i}^2 \le B \quad \text{for each } i = 1,\ldots,n.$

For instance, if the entries of $A$ take values in the range $[-1,1]$ , then (1) holds with $B = d$ . To sample a row $i$ with probability $\norm{a_i}^2/\norm{A}_{\rm F}^2$ , we perform the following the rejection sampling procedure:

Propose. Draw a random row index $1\le J\le n$ uniformly at random (i.e., $J$ is equally likely to be any row index between $1$ and $n$ .)
Accept/reject. Make a random choice: With probability $p_J = \norm{a_J}^2/B$ , accept and set $I\gets J$ . With the remaining probability $1-p_J$ , reject and return to step 1.

Why does this procedure work? As an intuitive explanation, notice that each row $i$ is equally likely to be proposed and has probability proportional to $\norm{a_i}^2$ of being accepted. Therefore, under this procedure, each row $i$ has probability proportional to $\norm{a_i}^2$ of being selected, just as desired for randomized Kaczmarz. If this intuitive justification is unsatisfying, we’ll have a formal derivation of correctness below (in a more general context).

I find rejection sampling to be super neat. In this case, it allows us to use simple ingredients (uniform row sampling and an upper bound on the row norms) to accomplish a more complicated task (sampling according to the squared row norms). We can sample from an interesting probability distribution (the squared row norm distribution) by thinning out samples from a simple distribution. This is a powerful idea that has many applications. (And, I believe, many more applications are waiting to be discovered.)

Rejection Sampling in General

Let’s now discuss rejection sampling in a bit more generality. Say I have a list of $n$ things each of which has a weight $w_i\ge 0$ . Our goal is to choose a random index $I$ with probability proportional to $w_i$ , i.e., $\prob \{ I = i\} = w_i / \sum_{j=1}^n w_j$ . We will call $w$ the target distribution. (Note that we do not require the weights $w_i$ to be normalized to satisfy $\sum_{i=1}^n w_i =1$ .)

Suppose that sampling from the target distribution is challenging, but we can sample from a proposal distribution $\rho$ , i.e., we can efficiently generate random $J$ such that $\prob \{J = i\} = \rho_i / \sum_{j=1}^n \rho_j$ . Further, suppose that the proposal distribution dominates the target distribution in the sense that

$w_i \le \rho_i \quad \text{for each } i=1,\ldots,n.$

Under this general setting, we can sample $I$ from the target distribution using rejection sampling, similar to above:

Propose. Draw a random row index $J$ from the proposal distribution.
Accept/reject. Make a random choice: With probability $p_J = w_J/\rho_J$ , accept and set $I\gets J$ . With the remaining probability $1-p_J$ , reject and return to step 1.

We recognize the squared row-norm sampling above as an example of this general strategy with $w_i = \norm{a_i}^2$ and $\rho_i = B$ for all $i$ .

Let us now convince ourselves more carefully that rejection sampling works. On any given execution of the rejection sampling loop, the probability of proposing index $i$ is $\rho_i / \sum_{j=1}^n \rho_j$ , after which the probability of accepting $i$ is $w_i / \rho_i$ . Therefore, the probability of outputting $i$ at any given loop is

$\prob \{\text{$i$ accepted this loop}\} = \frac{\rho_i}{\sum_{j=1}^n \rho_j} \cdot \frac{w_i}{\rho_i} = \frac{w_i}{\sum_{j=1}^n \rho_j}.$

The probability that any index $i$ on that loop is accepted is thus

$\prob \{\text{any accepted this loop}\} = \sum_{i=1}^n \prob \{\text{$i$ accepted this loop}\} = \frac{\sum_{i=1}^n w_i}{\sum_{i=1}^n \rho_i}.$

The rejection sampling loop accepts with fixed positive probability each execution, so it eventually accepts with 100% probability. Thus, the probability of accepting $i$ conditional on some index being accepted is

$\prob \{\text{$i$ accepted this loop} \mid \text{any accepted this loop}\} = \frac{\prob \{\text{$i$ accepted this loop}\}}{\prob \{\text{any accepted this loop}\}} = \frac{w_i}{\sum_{j=1}^n w_j}.$

Therefore, rejection sampling outputs a sample $I$ from the target distribution $w$ as promised.

As this analysis shows, the probability of accepting on any given execution of the rejection sampling loop is $\sum_{i=1}^n w_i / \sum_{i=1}^n \rho_i$ , the ratio of the total mass of the target $w_i$ to the total mass of the proposal $\rho_i$ . Thus, rejection sampling will have a high acceptance rate if $w_i \approx \rho_i$ and a low acceptance rate if $w_i \ll \rho_i$ . The conclusion for practice is that rejection sampling is only computationally efficient if one has access to a good proposal distribution $\rho_i$ , that is both easy to sample from and close to the target $w_i\approx \rho_i$ .

Here, we have presented rejection sampling as a strategy for sampling from a discrete set of items $i=1,\ldots,n$ , but this not all that rejection sampling can do. Indeed, one can also use rejection sampling for sampling a real-valued parameters $x \in \real$ or a multivariate parameter $x \in \real^n$ from a given (unnormalized) probability density function (or, if one likes, an unnormalized probability measure).

Before moving on, let us make one final observation. A very convenient thing about rejection sampling is that we only need the unnormalized probability weights $w_i$ and $\rho_i$ to implement the procedure, even though the total weights $\sum_{j=1}^n w_i$ and $\sum_{j=1}^n \rho_i$ are necessary to define the sampling probabilities $\prob \{I = i\} = w_i / \sum_{j=1}^n w_j$ and $\prob \{J = i\} = \rho_i / \sum_{j=1}^n \rho_j$ . This fact is necessary for many applications of rejection sampling. For instance, in the randomized Kaczmarz use, rejection sampling would be much less useful if we needed the total weight $\sum_{i=1}^n \norm{a_i}^2 = \norm{A}_{\rm F}^2$ , as computing $\norm{A}_{\rm F}^2$ requires a full pass over the data to compute.

Application: Randomly pivoted Cholesky

Another application of rejection sampling appears is to accelerate the randomly pivoted Cholesky algorithm, the subject of my recent paper. Here, I’ll provide an overview of the main idea with a focus on the role of rejection sampling in the procedure. See the paper for more details. Warning: Even as simplified here, this section presents a significantly more involved application of rejection sampling than we saw previously with randomized Kaczmarz!

Randomly pivoted Cholesky (RPCholesky) is a simple, but powerful algorithm for computing a low-rank approximation to a symmetric positive semidefinite matrix $A$ , and it can be used to accelerate kernel machine learning algorithms.

Conceptually, RPCholesky works as follows. Let $A^{(0)} \coloneqq A$ denote the initial matrix and $\hat{A}^{(0)} \coloneqq 0$ denote a trivial initial approximation. For $j=0,1,2,\ldots,k-1$ , RPCholesky performs the following steps:

Random sampling. Draw a random index $I_{j+1}$ with probability $\prob \{ I_{j+1} = i\} = A^{(j)}_{ii} / \sum_{i=1}^n A^{(j)}_{ii}$ .
Update approximation. Update $\hat{A}^{(j+1)} \gets \hat{A}^{(j)} + a_i^{(j)}(a_i^{(j)})^\top / A_{ii}^{(j)}$ . Here, $a_i^{(j)}$ denotes the $i$ th column of $A^{(j)}$ .
Update matrix. Update $A^{(j+1)} \gets A^{(j)} + a_i^{(j)}(a_i^{(j)})^\top / A_{ii}^{(j)}$ .

The result of this procedure is a rank- $k$ approximation $\hat{A}^{(k)}$ . With an optimized implementation, RPCholesky requires only $O(k^2N)$ operations and evaluates just $(k+1)N$ entries of the input matrix $A$ .

The standard RPCholesky algorithm is already fast, but there is room for improvement. The standard RPCholesky method processes the matrix $A$ one column $a_i$ at a time, but modern computers are faster at processing blocks of columns. Using the power of rejection sampling, we can develop faster versions of RPCholesky that take advantage of blocked computations. These blocked implementations are important: While we could handle a million-by-million matrix using the standard RPCholesky method, blocked implementations can be up to 40 $\times$ faster. In this post, I’ll describe the simplest way of using rejection sampling to implement RPCholesky.

To set the stage, let’s first establish some notation and make a few observations. Let $S_j = \{I_1,\ldots,I_j\}$ denote the first $j$ selected random indices. With a little matrix algebra, one can show that the approximation $\hat{A}^{(j)}$ produced by $j$ steps of RPCholesky obeys the formula¹

(1) $\hat{A}^{(j)} = A(:,S_j) A(S_j,S_j)^{-1}A(S_j,:).$

Here, we are using MATLAB notation for indexing the matrix $A$ . The residual matrix $A^{(j)}$ is given by

$A^{(j)} = A - \hat{A}^{(j)} = A - A(:,S_j) A(S_j,S_j)^{-1}A(S_j,:).$

Importantly for us, observe that we can evaluate the $i$ th diagonal entry of $A^{(j)}$ using the formula

(2) $A^{(j)}_{ii} = A_{ii} - A(i,S_j) A(S_j,S_j)^{-1} A(S_j,i).$

Compared to operating on the full input matrix $A$ , evaluating an entry $A^{(j)}_{ii}$ is cheap, just requiring some arithmetic involving matrices and vectors of size $j$ .

Here, we see a situation similar to randomized Kaczmarz. At each step of RPCholesky, we must sample a random index $I_{j+1}$ from a target distribution $w_i = A_{ii}^{(j)}$ , but it is expensive to evaluate all of the $w_i$ ‘s. This is a natural setting for rejection sampling. As proposal distribution, we may use the initial diagonal of $A$ , $\rho_i = A_{ii}$ . (One may verify that, as required, $w_i = A_{ii}^{(j)} \le A_{ii} = \rho_i$ for all $i$ .) This leads to a rejection sampling implementation for RPCholesky:

Sample indices. For $j=0,1,\ldots,k-1$ , sample random indices $I_{j+1}$ from the target distribution $w_i = A^{(j)}_{ii}$ using rejection sampling with proposal distribution $\rho_i = A_{ii}$ . Entries $A_{ii}^{(j)}$ are evaluated on an as-needed basis using (2).
Build the approximation. Form the approximation $\hat{A}^{(k)}$ all at once using (1).

By using rejection sampling, we have gone from an algorithm which handles the matrix one column at a time to a new algorithm which processes columns of the matrix $A$ all at once through the formula (1). In the right situations, this new algorithm can be significantly faster than the original RPCholesky method.²

We have now seen two extreme cases: the standard RPCholesky algorithm that processes the columns of $A$ one at a time and a rejection sampling implementation that operates on the columns of $A$ all at once. In practice, the fastest algorithm sits between these extremes, using rejection sampling to sample column indices in moderate-size blocks (say, between 100 and 1000 columns at a time). This “goldilocks” algorithm is the accelerated RPCholesky method that we introduce in our paper. Check it out for details!

Conclusion: Rejection Sampling, an Algorithm Design Tool

We’ve now seen two applications, randomized Kaczmarz and randomly pivoted Cholesky, where rejection sampling can be used to speed up an algorithm. In both cases, we wanted to sample from a distribution over $n$ row or column indices, but it was expensive to perform a census to compute the probability weight of each item individually. Rejection sampling offers a different approach, where we generate samples from a cheap-to-sample proposal distribution and only evaluate the probability weights of the proposed items. In the right situations, this rejection sampling approach can lead to significant computational speedups.

Rejection sampling has been used by statistics-oriented folks for decades, but the power of this technique seems less well-known to researchers in applied math, scientific computation, and related areas. I think there are a lot more exciting ways that we can use rejection sampling to design better, faster randomized algorithms.

Neat Randomized Algorithms: RandDiag for Rapidly Diagonalizing Normal Matrices

July 2, 2024 by Ethan N. Epperly Leave a comment

Consider two complex-valued square matrices $A\in\complex^{n\times n}$ and $B\in\complex^{n\times n}$ . The first matrix $A$ is Hermitian, being equal $A = A^*$ to its conjugate transpose $A^*$ . The other matrix $B$ is non-Hermitian, $B \ne B^*$ . Let’s see how long it takes to compute their eigenvalue decompositions in MATLAB:

>> A = randn(1e3) + 1i*randn(1e3); A = (A+A')/2;
>> tic; [V_A,D_A] = eig(A); toc % Hermitian matrix
Elapsed time is 0.415145 seconds.
>> B = randn(1e3) + 1i*randn(1e3);
>> tic; [V_B,D_B] = eig(B); toc % non-Hermitian matrix
Elapsed time is 1.668246 seconds.

We see that it takes $4\times$ longer to compute the eigenvalues of the non-Hermitian matrix $B$ as compared to the Hermitian matrix $A$ . Moreover, the matrix $V_A$ of eigenvectors for a Hermitian matrix $A = V_AD_AV_A^{-1}$ is a unitary matrix, $V_A^*V_A = V_AV_A^* = I$ .

There are another class of matrices with nice eigenvalue decompositions, normal matrices. A square, complex-valued matrix $C$ is normal if $C^*C = CC^*$ . The matrix $V_C$ of eigenvectors for a normal matrix $C = V_C D_C V_C^{-1}$ is also unitary, $V_C^*V_C = V_CV_C^* = I$ . An important class of normal matrices are unitary matrices themselves. A unitary matrix $U$ is always normal since it satisfies $U^*U = UU^* = I$ .

Let’s see how long it takes MATLAB to compute the eigenvalue decomposition of a unitary (and thus normal) matrix:

>> U = V_A;                     % unitary, and thus normal, matrix
>> tic; [V_U,D_U] = eig(U); toc % normal matrix
Elapsed time is 2.201017 seconds.

Even longer than it took to compute an eigenvalue decomposition of the non-normal matrix $B$ ! Can we make the normal eigenvalue decomposition closer to the speed of the Hermitian eigenvalue decomposition?

Here is the start of an idea. Every square matrix $C$ has a Cartesian decomposition:

$C = H + iS, \quad H = \frac{C+C^*}{2}, \quad S = \frac{C-C^*}{2i}.$

We have written $C$ as a combination of its Hermitian part $H$ and $i$ times its skew-Hermitian part $S$ . Both $H$ and $S$ are Hermitian matrices. The Cartesian decomposition of a square matrix is analogous to the decomposition of a complex number into its real and imaginary parts.

For a normal matrix $C$ , the Hermitian and skew-Hermitian parts commute, $HS = SH$ . We know from matrix theory that commuting Hermitian matrices are simultaneously diagonalizable, i.e., there exists $Q$ such that $H = QD_HQ^*$ and $S = QD_SQ^*$ for diagonal matrices $D_H$ and $D_S$ . Thus, given access to such $Q$ , $C$ has eigenvalue decomposition

$C = Q(D_H+iD_S)Q^*.$

Here’s a first attempt to turn this insight into an algorithm. First, compute the Hermitian part $H$ of $C$ , diagonalize $H = QD_HQ^*$ , and then see if $Q$ diagonalizes $C$ . Let’s test this out on a $2\times 2$ example:

>> C = orth(randn(2) + 1i*randn(2)); % unitary matrix
>> H = (C+C')/2;                     % Hermitian part
>> [Q,~] = eig(H);
>> Q'*C*Q                            % check to see if diagonal
ans =
  -0.9933 + 0.1152i  -0.0000 + 0.0000i
   0.0000 + 0.0000i  -0.3175 - 0.9483i

Yay! We’ve succeeded at diagonalizing the matrix $C$ using only a Hermitian eigenvalue decomposition. But we should be careful about declaring victory too early. Here’s a bad example:

>> C = [1 1i;1i 1]; % normal matrix
>> H = (C+C')/2;
>> [Q,~] = eig(H);
>> Q'*C*Q           % oh no! not diagonal
ans =
   1.0000 + 0.0000i   0.0000 + 1.0000i
   0.0000 + 1.0000i   1.0000 + 0.0000i

What’s going on here? The issue is that the Hermitian part $H = I$ for this matrix has a repeated eigenvalue. Thus, $H$ has multiple different valid matrices of eigenvectors. (In this specific case, every unitary matrix $Q$ diagonalizes $H$ .) By looking at $H$ alone, we don’t know which $Q$ matrix to pick which also diagonalizes $S$ .

He and Kressner developed a beautifully simple randomized algorithm called RandDiag to circumvent this failure mode. The idea is straightforward:

Form a random linear combination $M = \gamma_1 H + \gamma_2 S$ of the Hermitian and skew-Hermitian parts of $A$ , with standard normal random coefficients $\gamma_1$ and $\gamma_2$ .
Compute $Q$ that diagonalizes $M$ .

That’s it!

To get a sense of why He and Kressner’s algorithm works, suppose that $H$ has some repeated eigenvalues and $S$ has all distinct eigenvalues. Given this setup, it seems likely that a random linear combination of $S$ and $H$ will also have all distinct eigenvalues. (It would take a very special circumstances for a random linear combination to yield two eigenvalues that are exactly the same!) Indeed, this intuition is a fact: With 100% probability, $Q$ diagonalizing a Gaussian random linear combination of simultaneously diagonalizable matrices $H$ and $S$ also diagonalizes $H$ and $S$ individually.

MATLAB code for RandDiag is as follows:

function Q = rand_diag(C)
   H = (C+C')/2; S = (C-C')/2i;
   M = randn*H + randn*S;
   [Q,~] = eig(M);
end

When applied to our hard $2\times 2$ example from before, RandDiag succeeds at giving a matrix that diagonalizes $C$ :

>> Q = rand_diag(C);
>> Q'*C*Q
ans =
   1.0000 - 1.0000i  -0.0000 + 0.0000i
  -0.0000 - 0.0000i   1.0000 + 1.0000i

For computing the matrix of eigenvectors for a $1000\times 1000$ unitary matrix, RandDiag takes 0.4 seconds, just as fast as the Hermitian eigendecomposition did.

>> tic; V_U = rand_diag(U); toc
Elapsed time is 0.437309 seconds.

He and Kressner’s algorithm is delightful. Ultimately, it uses randomness in only a small way. For most coefficients $a_1,a_2 \in \real$ , a matrix $Q$ diagonalizing $a_1 H + a_2 S$ will also diagonalize $A = H+iS$ . But, for any specific choice of $a_1,a_2$ , there is a possibility of failure. To avoid this possibility, we can just pick $a_1$ and $a_2$ at random. It’s really as simple as that.

References: RandDiag was proposed in A simple, randomized algorithm for diagonalizing normal matrices by He and Kressner (2024), building on their earlier work in Randomized Joint Diagonalization of Symmetric Matrices (2022) which considers the general case of using random linear combinations to (approximately) simultaneous diagonalize (nearly) commuting matrices. RandDiag is an example of a linear algebraic algorithm that uses randomness to put the input into “general position”; see Randomized matrix computations: Themes and variations by Kireeva and Tropp (2024) for a discussion of this, and other, ways of using randomness to design matrix algorithms.

Neat Randomized Algorithms: Randomized Cholesky QR

June 25, 2024 by Ethan N. Epperly Leave a comment

As a research area, randomized numerical linear algebra (RNLA) is as hot as ever. To celebrate the exciting work in this space, I’m starting a new series on my blog where I celebrate cool recent algorithms in the area. At some future point, I might talk about my own work in this series, but for now I’m hoping to use this series to highlight some of the awesome work being done by my colleagues.

Given a tall matrix $A \in \real^{m\times n}$ with $m\ge n$ , its (economy-size) QR factorization is a decomposition of the form $A = QR$ , where $Q \in \real^{m\times n}$ is a matrix with orthonormal columns and $R \in \real^{n\times n}$ is upper triangular. QR factorizations are used to solve least-squares problems and as a computational procedure to orthonormalize the columns of a matrix.

Here’s an example in MATLAB, where we use QR factorization to orthonormalize the columns of a $10^6\times 10^2$ test matrix. It takes about 2.5 seconds to run.

>> A = randn(1e6, 1e2) * randn(1e2) * randn(1e2); % test matrix
>> tic; [Q,R] = qr(A,"econ"); toc
Elapsed time is 2.647317 seconds.

The classical algorithm for computing a QR factorization uses Householder reflectors and is exceptionally numerically stable. Since $Q$ has orthonormal columns, $Q^\top Q = I$ is the identity matrix. Indeed, this relation holds up to a tiny error for the $Q$ computed by Householder QR:

>> norm(Q'*Q - eye(1e2)) % || Q^T Q - I ||
ans =
   7.0396e-14

The relative error $\|A - QR\|/\|A\|$ is also small:

>> norm(A - Q*R) / norm(A)
ans =
   4.8981e-14

Here is an alternate procedure for computing a QR factorization, known as Cholesky QR:

function [Q,R] = cholesky_qr(A)
   R = chol(A'*A);
   Q = A / R;       % Q = A * R^{-1}
end

This algorithm works by forming $A^\top A$ , computing its (upper triangular) Cholesky decomposition $A^\top A = R^\top R$ , and setting $Q = AR^{-1}$ . Cholesky QR is very fast, about $5\times$ faster than Householder QR for this example:

>> tic; [Q,R] = cholesky_qr(A); toc
Elapsed time is 0.536694 seconds.

Unfortunately, Cholesky QR is much less accurate and numerically stable than Householder QR. Here, for instance, is the value of $\|Q^\top Q - I\|$ , about ten million times larger than for Householder QR!:

>> norm(Q'*Q - eye(1e2))
ans =
   7.5929e-07

What’s going on? As we’ve discussed before on this blog, forming $A^\top A$ is typically problematic in linear algebraic computations. The “badness” of a matrix $A$ is measured by its condition number, defined to be the ratio of its largest and smallest singular values $\kappa(A) = \sigma_{\rm max}(A)/\sigma_{\rm min}(A)$ . The condition number of $A^\top A$ is the square of the condition number of $A$ , $\kappa(A^\top A) = \kappa(A)^2$ , which is at the root of Cholesky QR’s loss of accuracy. Thus, Cholesky QR is only appropriate for matrices that are well-conditioned, having a small condition number $\kappa(A) \approx 1$ , say $\kappa(A) \le 10$ .

The idea of randomized Cholesky QR is to use randomness to precondition $A$ , producing a matrix $R_1$ that $B = AR_1^{-1}$ is well-conditioned. Then, since $B$ is well-conditioned, we can apply ordinary Cholesky QR to it without issue. Here are the steps:

Draw a sketching matrix $S$ of size $2n\times m$ ; see these posts of mine for an introduction to sketching.
Form the sketch $SA$ . This step compresses the very tall matrix $m\times n$ to the much shorter matrix $SA$ of size $2n\times n$ .
Compute a QR factorization $SA = Q_1R_1$ using Householder QR. Since the matrix $SA$ is small, this factorization will be quick to compute.
Form the preconditioned matrix $B = AR_1^{-1}$ .
Apply Cholesky QR to $B$ to compute $B = QR_2$ .
Set $R = R_2R_1$ . Observe that $A = BR_1 = QR_2R_1 = QR$ , as desired.

MATLAB code for randomized Cholesky QR is provided below:¹

function [Q,R] = rand_cholesky_qr(A)
   S = sparsesign(2*size(A,2),size(A,1),8); % sparse sign embedding
   R1 = qr(S*A,"econ"); % sketch and (Householder) QR factorize
   B = A / R1; % B = A * R_1^{-1}
   [Q,R2] = cholesky_qr(B);
   R = R2*R1;
end

Randomized Cholesky QR is still faster than ordinary Householder QR, about $3\times$ faster in our experiment:

>> tic; [Q,R] = rand_cholesky_qr(A); toc
Elapsed time is 0.920787 seconds.

Randomized Cholesky QR greatly improves on ordinary Cholesky QR in terms of accuracy and numerical stability. In fact, the size of $\|Q^\top Q - I\|$ is even smaller than for Householder QR!

>> norm(Q'*Q - eye(1e2))
ans =
   1.0926e-14

The relative error $\|A - QR\|/\|A\|$ is small, too! Even smaller than for Householder QR in fact:

>> norm(A - Q*R) / norm(A)
ans =
   4.0007e-16

Like many great ideas, randomized Cholesky QR was developed independently by a number of research groups. A version of this algorithm was first introduced in 2021 by Fan, Guo, and Lin. Similar algorithms were investigated in 2022 and 2023 by Balabanov, Higgins et al., and Melnichenko et al. Check out Melnichenko et al.‘s paper in particular, which shows very impressive results for using randomized Cholesky QR to compute column pivoted QR factorizations.

References: Primary references are A Novel Randomized XR-Based Preconditioned CholeskyQR Algorithm by Fan, Guo, and Lin (2021); Randomized Cholesky QR factorizations by Balabanov (2022); Analysis of Randomized Householder-Cholesky QR Factorization with Multisketching by Higgins et al. (2023); CholeskyQR with Randomization and Pivoting for Tall Matrices (CQRRPT) by Melnichenko et al. (2023). The idea of using sketching to precondition tall matrices originates in the paper A fast randomized algorithm for overdetermined linear least-squares regression by Rokhlin and Tygert (2008).

Don’t Use Gaussians in Stochastic Trace Estimation

January 28, 2024 by Ethan N. Epperly 1 Comment

Suppose we are interested in estimating the trace $\tr(A) = \sum_{i=1}^n A_{ii}$ of an $n\times n$ matrix $A$ that can be only accessed through matrix–vector products $Ax_1,\ldots,Ax_m$ . The classical method for this purpose is the Girard–Hutchinson estimator

$\hat{\tr} = \frac{1}{m} \left( x_1^\top Ax_1 + \cdots + x_m^\top Ax_m \right),$

where the vectors $x_1,\ldots,x_m$ are independent, identically distributed (iid) random vectors satisfying the isotropy condition

$\expect[x_ix_i^\top] = I.$

Examples of vectors satisfying this condition include

Gaussian: The entries of each $x_i$ are iid standard Gaussian random variables.
Random signs: The entries of each $x_i$ are iid random numbers taking the values $\pm 1$ with equal probabilities (i.e., Rademacher random variables).
Sphere: Each $x_i$ is a uniformly random point on the (Euclidean) sphere of radius $\sqrt{n}$ , $x_i \sim \operatorname{Unif} \{ y \in \mathbb{R}^n : y^\top y = n \}$ .

Stochastic trace estimation has a number of applications: log-determinant computations in machine learning, partition function calculations in statistical physics, generalized cross validation for smoothing splines, and triangle counting in large networks. Several improvements to the basic Girard–Hutchinson estimator have been developed recently. I am partial to XTrace, an improved trace estimator that I developed with my collaborators.

This post is addressed at the question:

Which distribution should be used for the test vectors $x_i$ for stochastic trace estimation?

Since the Girard–Hutchinson estimator is unbiased $\expect[\hat{\tr}] = \tr(A)$ , the variance of $\hat{\tr}$ is equal to the mean-square error. Thus, the lowest variance trace estimate is the most accurate. In my previous post on trace estimation, I discussed formulas for the variance $\Var(\hat{\tr})$ of the Girard–Hutchinson estimator with different choices of test vectors. In that post, I stated the formulas for different choices of test vectors (Gaussian, random signs, sphere) and showed how those formulas could be proven.

In this post, I will take the opportunity to editorialize on which distribution to pick. The thesis of this post is as follows:

The sphere distribution is essentially always preferable to the Gaussian distribution for trace estimation.

To explain why, let’s focus on the case when $A$ is real and symmetric.¹ Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $A$ and define the eigenvalue mean

$\overline{\lambda} = \frac{\lambda_1 + \cdots + \lambda_n}{n}.$

Then the variance of the Girard–Hutchinson estimator with Gaussian vectors $x_i$ is

$\Var(\hat{\tr}_{\rm Gaussian}) = \frac{1}{m} \cdot 2 \sum_{i=1}^n \lambda_i^2.$

For vectors $x_i$ drawn from the sphere, we have

$\Var(\hat{\tr}_{\rm sphere}) = \frac{1}{m} \cdot \frac{n}{n+2} \cdot 2\sum_{i=1}^n (\lambda_i - \overline{\lambda})^2.$

The sphere distribution improves on the Gaussian distribution in two ways. First, the variance of $\Var(\hat{\tr}_{\rm sphere})$ is smaller than $\Var(\hat{\tr}_{\rm Gaussian})$ by a factor of $n/(n+2) < 1$ . This improvement is quite minor. Second, and more importantly, $\Var(\hat{\tr}_{\rm Gaussian})$ is proportional to the sum of $A$ ‘s squared eigenvalues whereas $\Var(\hat{\tr}_{\rm sphere})$ is proportional to the sum of $A$ ‘s squared eigenvalues after having been shifted to be mean-zero!

The difference between Gaussian and sphere test vectors can be large. To see this, consider a $1000\times 1000$ matrix $A$ with eigenvalues uniformly distributed between $0.9$ and $1.1$ with a (Haar orthgonal) random matrix of eigenvectors. For simplicity, since the variance of all Girard–Hutchinson estimates is proportional to $1/m$ , we take $m=1$ . Below show the variance of Girard–Hutchinson estimator for different distributions for the test vector. We see that the sphere distribution leads to a trace estimate which has a variance 300× smaller than the Gaussian distribution. For this example, the sphere and random sign distributions are similar.

Which Distribution Should You Use: Signs vs. Sphere

The main point of this post is to argue against using the Gaussian distribution. But which distribution should you use: Random signs? The sphere distribution? The answer, for most applications, is one of those two, but exactly which depends on the properties of the matrix $A$ .

The variance of the Girard–Hutchinson estimator with the random signs estimator is

$\Var(\hat{\tr}_{\rm signs}) = 2 \sum_{i\ne j} A_{ij}^2.$

Thus, $\Var(\hat{\tr}_{\rm signs})$ depends on the size of the off-diagonal entries of $A$ ; $\Var(\hat{\tr}_{\rm signs})$ does not depend on the diagonal of $A$ at all! For matrices with small off-diagonal entries (such as diagonally dominant matrices), the random signs distribution is often the best.

However, for other problems, the sphere distribution is preferable to random signs. The sphere distribution is rotation-invariant, so $\Var(\hat{\tr}_{\rm sphere})$ is independent of the eigenvectors of the (symmetric) matrix $A$ , depending only on $A$ ‘s eigenvalues. By contrast, the variance of the Girard–Hutchinson estimator with the random signs distribution can significantly depend on the eigenvectors of the matrix $A$ . For a given set of eigenvalues and the worst-case choice of eigenvectors, $\Var(\hat{\tr}_{\rm sphere})$ will always be smaller than $\Var(\hat{\tr}_{\rm signs})$ . In fact, $\Var(\hat{\tr}_{\rm sphere})$ is the minimum variance distribution for Girard–Hutchinson trace estimation for a matrix with fixed eigenvalues and worst-case eigenvectors; see this section of my previous post for details.

In my experience, random signs and the sphere distribution are both perfectly adequate for trace estimation and either is a sensible default if you’re developing software. The Gaussian distribution on the other hand… don’t use it unless you have a good reason to.

How Good Can Stochastic Trace Estimates Be?

January 4, 2024 by Ethan N. Epperly 1 Comment

I am excited to share that our paper XTrace: Making the most of every sample in stochastic trace estimation has been published in the SIAM Journal on Matrix Analysis and Applications. (See also our paper on arXiv.)

Spurred by this exciting news, I wanted to take the opportunity to share one of my favorite results in randomized numerical linear algebra: a “speed limit” result of Meyer, Musco, Musco, and Woodruff that establishes a fundamental limitation on how accurate any trace estimation algorithm can be.

Let’s back up. Given an unknown square matrix $A$ , the trace of $A$ , defined to be the sum of its diagonal entries

$\tr(A) \coloneqq \sum_{i=1}^n A_{ii}.$

The catch? We assume that we can only access the matrix $A$ through matrix–vector products (affectionately known as “matvecs”): Given any vector $x$ , we have access to $Ax$ . Our goal is to form an estimate $\hat{\tr}$ that is as accurate as possible while using as few matvecs as we can get away with.

To simplify things, let’s assume the matrix $A$ is symmetric and positive (semi)definite. The classical algorithm for trace estimation is due to Gir ard and Hutchinson, producing a probabilistic estimate $\hat{\tr}$ with a small average (relative) error:

$\expect\left[\frac{|\hat{\tr}-\tr(A)|}{\tr(A)}\right] \le \varepsilon \quad \text{using } m= \frac{\rm const}{\varepsilon^2} \text{ matvecs}.$

If one wants high accuracy, this algorithm is expensive. To achieve just a 1% error ( $\varepsilon=0.01$ ) requires roughly $m=10,\!000$ matvecs!

This state of affairs was greatly improved by Meyer, Musco, Musco, and Woodruff. Building upon previous work, they proposed the Hutch++ algorithm and proved it outputs an estimate $\hat{\tr}$ satisfying the following bound:

(1) $\expect\left[\frac{|\hat{\tr}-\tr(A)|}{\tr(A)}\right] \le \varepsilon \quad \text{using } m= \frac{\rm const}{\varepsilon} \text{ matvecs}.$

Now, we only require roughly $m=100$ matvecs to achieve 1% error! Our algorithm, XTrace, satisfies the same error guarantee (1) as Hutch++. On certain problems, XTrace can be quite a bit more accurate than Hutch++.

The MMMW Trace Estimation “Speed Limit”

Given the dramatic improvement of Hutch++ and XTrace over Girard–Hutchinson, it is natural to hope: Is there an algorithm that does even better than Hutch++ and XTrace? For instance, is there an algorithm satisfying an even slightly better error bound of the form

$\expect\left[\frac{|\hat{\tr}-\tr(A)|}{\tr(A)}\right] \le \varepsilon \quad \text{using } m= \frac{\rm const}{\varepsilon^{0.999}} \text{ matvecs}?$

Unfortunately not. Hutch++ and XTrace are essentially as good as it gets.

Let’s add some fine print. Consider an algorithm for the trace estimation problem. Whenever the algorithm wants, it can present a vector $x_i$ and receive back $Ax_i$ . The algorithm is allowed to be adaptive: It can use the matvecs $Ax_1,\ldots,Ax_s$ it has already collected to decide which vector $x_{s+1}$ to present next. We measure the cost of the algorithm in terms of the number of matvecs alone, and the algorithm knows nothing about the psd matrix $A$ other what it learns from matvecs.

One final stipulation:

Simple entries assumption. We assume that the entries of the vectors $x_i$ presented by the algorithm are real numbers between $-1$ and $1$ with up to $b$ digits after the decimal place.

To get a feel for this simple entries assumption, suppose we set $b=2$ . Then $(-0.92,0.17)$ would be an allowed input vector, but $(0.232,-0.125)$ would not be (too many digits after the decimal place). Similarly, $(18.3,2.4)$ would not be valid because its entries exceed $1$ . The simple entries assumption is reasonable as we typically represent numbers on digital computers by storing a fixed number of digits of accuracy.¹

With all these stipulations, we are ready to state the “speed limit” for trace estimation proved by Meyer, Musco, Musco, and Woodruff:

Informal theorem (Meyer, Musco, Musco, Woodruff). Under the assumptions above, there is no trace estimation algorithm producing an estimate $\hat{\tr}$ satisfying
$\expect\left[\frac{|\hat{\tr}-\tr(A)|}{\tr(A)}\right] \le \varepsilon \quad \text{using } m= \frac{\rm const}{\varepsilon^{0.999}} \text{ matvecs}.$

We will see a slightly sharper version of the theorem below, but this statement captures the essence of the result.

Communication Complexity

To prove the MMMW theorem, we have to take a journey to the beautiful subject of communication complexity. The story is this. Alice and Bob are interested in solving a computational problem together. Alice has her input $x$ and Bob has his input $y$ , and they are interested in computing a function $f(x,y)$ of both their inputs.

Unfortunately for the two of them, Alice and Bob are separated by a great distance, and can only communicate by sending single bits (0 or 1) of information over a slow network connection. Every bit of communication is costly. The field of communication complexity is dedicated to determining how efficiently Alice and Bob are able to solve problems of this form.

The Gap-Hamming problem is one example of a problem studied in communication complexity. As inputs, Alice and Bob receive vectors $x,y \in \{\pm 1\}^n$ with $+1$ and $-1$ entries from a third party Eve. Eve promises Alice and Bob that their vectors $x$ and $y$ satisfy one of two conditions:

(2) $\text{Case 0: } x^\top y \ge\sqrt{n} \quad \text{or} \quad \text{Case 1: } x^\top y \le -\sqrt{n}.$

Alice and Bob must work together, sending as few bits of communication as possible, to determine which case they are in.

There’s one simple solution to this problem: First, Bob sends his whole input vector $y$ to Alice. Each entry of $y$ takes one of the two value $\pm 1$ and can therefore be communicated in a single bit. Having received $y$ , Alice computes $x^\top y$ , determines whether they are in case 0 or case 1, and sends Bob a single bit to communicate the answer. This procedure requires $n+1$ bits of communication.

Can Alice and Bob still solve this problem with many fewer than $n$ bits of communication, say $\sqrt{n}$ bits? Unfortunately not. The following theorem of Chakrabati and Regev shows that roughly $n$ bits of communication are needed to solve this problem:

Theorem (Chakrabati–Regev). Any algorithm which solves the Gap-Hamming problem that succeeds with at least $2/3$ probability for every pair of inputs $x$ and $y$ (satisfying one of the conditions (2)) must take $\Omega(n)$ bits of communication.

Here, $\Omega(n)$ is big-Omega notation, closely related to big-O notation $\order(n)$ and big-Theta notation $\Theta(n)$ . For the less familiar, it can be helpful to interpret $\Omega(n)$ , $\order(n)$ , and $\Theta(n)$ as all standing for “proportional to $n$ ”. In plain language, the theorem of Chakrabati and Regev result states that there is no algorithm for the Gap-Hamming problem that much more effective than the basic algorithm where Bob sends his whole input to Alice (in the sense of requiring less than $\order(n)$ bits of communication).

Reducing Gap-Hamming to Trace Estimation

This whole state of affairs is very sad for Alice and Bob, but what does it have to do with trace estimation? Remarkably, we can use hardness of the Gap-Hamming problem to show there’s no algorithm that fundamentally improves on Hutch++ and XTrace. The argument goes something like this:

If there were a trace estimation algorithm fundamentally better than Hutch++ and XTrace, we could use it to solve Gap-Hamming in fewer than $\order(n)$ bits of communication.
But no algorithm can solve Gap-Hamming in fewer than $\order(n)$ bits or communication.
Therefore, no trace estimation algorithm is fundamentally better than Hutch++ and XTrace.

Step 2 is the work of Chakrabati and Regev, and step 3 follows logically from 1 and 2. Therefore, we are left to complete step 1 of the argument.

Protocol

Assume we have access to a really good trace estimation algorithm. We will use it to solve the Gap-Hamming problem. For simplicity, assume $n$ is a perfect square. The basic idea is this:

Have Alice and Bob reshape their inputs $x,y \in \{\pm 1\}^n$ into matrices $X,Y\in\{\pm 1\}^{\sqrt{n}\times \sqrt{n}}$ , and consider (but do not form!) the positive semidefinite matrix
$A = (X+Y)^\top (X+Y).$
Observe that
$\tr(A) = \tr(X^\top X) + 2\tr(X^\top Y) + \tr(Y^\top Y) = 2n + 2(x^\top y).$
Thus, the two cases in (2) can be equivalently written in terms of $\tr(A)$ :
(2′) $\text{Case 0: } \tr(A)\ge 2n + 2\sqrt{n} \quad \text{or} \quad \text{Case 1: } \tr(A) \le 2n-2\sqrt{n}.$
By working together, Alice and Bob can implement a trace estimation algorithm. Alice will be in charge of running the algorithm, but Alice and Bob must work together to compute matvecs. (Details below!)
Using the output of the trace estimation algorithm, Alice determines whether they are in case 0 or 1 (i.e., where $\tr(A) \gg 2n$ or $\tr(A) \ll 2n$ ) and sends the result to Bob.

To complete this procedure, we just need to show how Alice and Bob can implement the matvec procedure using minimal communication. Suppose Alice and Bob want to compute $Az$ for some vector $z$ with entries between $-1$ and $1$ with up to $b$ decimal digits. First, convert $z$ to a vector $w\coloneqq 10^b z$ whose entries are integers between $-10^b$ and $10^b$ . Since $Az = 10^{-b}Aw$ , interconverting between $Az$ and $Aw$ is trivial. Alice and Bob’s procedure for computing $Aw$ is as follows:

Alice sends Bob $w$ .
Having received $w$ , Bob forms $Yw$ and sends it to Alice.
Having received $Yw$ , Alice computes $v\coloneqq Xw+Yw$ and sends it to Bob.
Having received $v$ , Bob computes $Y^\top v$ and sends its to Alice.
Alice forms $Aw = X^\top v + Y^\top v$ .

Because $X$ and $Y$ are $\sqrt{n}\times \sqrt{n}$ and have $\pm 1$ entries, all vectors computed in this procedure are vectors of length $\sqrt{n}$ with integer entries between $-4n 10^b$ and $4n10^b$ . We conclude the communication cost for one matvec is $T\coloneqq\Theta((b+\log n)\sqrt{n})$ bits.

Analysis

Consider an algorithm we’ll call BestTraceAlgorithm. Given any accuracy parameter $\varepsilon > 0$ , BestTraceAlgorithm requires at most $m = m(\varepsilon)$ matvecs and, for any positive semidefinite input matrix $A$ of any size, produces an estimate $\hat{\tr}$ satisfying

(3) $\expect\left[\frac{|\hat{\tr}-\tr(A)|}{\tr(A)}\right] \le \varepsilon.$

We assume that BestTraceAlgorithm is the best possible algorithm in the sense that no algorithm can achieve (3) on all (positive semidefinite) inputs with $m' < m$ matvecs.

To solve the Gap-Hamming problem, Alice and Bob just need enough accuracy in their trace estimation to distinguish between cases 0 and 1. In particular, if

$\left| \frac{\hat{\tr} - \tr(A)}{\tr(A)} \right| \le \frac{1}{\sqrt{n}},$

then Alice and Bob can distinguish between cases 0 and 1 in (2′)

Suppose that Alice and Bob apply trace estimation to solve the Gap-Hamming problem, using $m$ matvecs in total. The total communication is $m\cdot T = \order(m(b+\log n)\sqrt{n})$ bits. Chakrabati and Regev showed that Gap-Hamming requires $cn$ bits of communication (for some $c>0$ ) to solve the Gap-Hamming problem with $2/3$ probability. Thus, if $m\cdot T < cn$ , then Alice and Bob fail to solve the Gap-Hamming problem with at least $1/3$ probability. Thus,

$\text{If } m < \frac{cn}{T} = \Theta\left( \frac{\sqrt{n}}{b+\log n} \right), \quad \text{then } \left| \frac{\hat{\tr} - \tr(A)}{\tr(A)} \right| > \frac{1}{\sqrt{n}} \text{ with probability at least } \frac{1}{3}.$

The contrapositive of this statement is that if

$\text{If }\left| \frac{\hat{\tr} - \tr(A)}{\tr(A)} \right| \le \frac{1}{\sqrt{n}}\text{ with probability at least } \frac{2}{3}, \quad \text{then } m \ge \Theta\left( \frac{\sqrt{n}}{b+\log n} \right).$

Say Alice and Bob run BestTraceAlgorithm with parameter $\varepsilon = \tfrac{1}{3\sqrt{n}}$ . Then, by (3) and Markov’s inequality,

$\left| \frac{\hat{\tr} - \tr(A)}{\tr(A)} \right| \le \frac{1}{\sqrt{n}} \quad \text{with probability at least }\frac{2}{3}.$

Therefore, BestTraceAlgorithm requires at least

$m \ge \Theta\left( \frac{\sqrt{n}}{b+\log n} \right) \text{ matvecs}.$

Using the fact that we’ve set $\varepsilon = 1/3\sqrt{n}$ , we conclude that any trace estimation algorithm, even BestTraceAlgorithm, requires

$m \ge \Theta \left( \frac{1}{\varepsilon (b+\log(1/\varepsilon))} \right) \text{ matvecs}.$

In particular, no trace estimation algorithm can achieve mean relative error $\varepsilon$ using even $\order(1/\varepsilon^{0.999})$ matvecs. This proves the MMMW theorem.

Five Interpretations of Kernel Quadrature

December 1, 2023 by Ethan N. Epperly Leave a comment

I’m excited to share that my paper Kernel quadrature with randomly pivoted Cholesky, joint with Elvira Moreno, has been accepted to NeurIPS 2023 as a spotlight.

Today, I want to share with you a little about the kernel quadrature problem. To avoid this post getting too long, I’m going to write this post assuming familiarity with the concepts of reproducing kernel Hilbert spaces and Gaussian processes.

Integration and Quadrature

Integration is one of the most widely used operations in mathematics and its applications. As such, it is a basic problem of wide interest to develop numerical methods for evaluating integrals.

In this post, we will consider a quite general integration problem. Let $\Omega\subseteq \real^d$ be a domain and let $\mu$ be a (finite Borel) measure on $\Omega$ . We consider the task of evaluating

$I[f] = \int_\Omega f(x) g(x) \, \mathrm{d}\mu(x).$

One can imagine that $g$ , $\mu$ , and $\Omega$ are fixed, but we may want to evaluate this same integral $I[f]$ for multiple different functions $f$ .

To evaluate, we will design a quadrature approximation to the integral $I[f]$ :

$\hat{I}_{w,s}[f] = \sum_{i=1}^n w_i f(s_i) \approx I[f].$

Concretely, we wish to find real numbers $w = (w_1,\ldots,w_n) \in \real^n$ and points $s = (s_1,\ldots,s_n) \in \Omega^n$ such that the approximation $\hat{I}_{w,s}[f] \approx I[f]$ is accurate.

Smoothness and Reproducing Kernel Hilbert Spaces

As is frequently the case in computational mathematics, the accuracy we can expect for this integration problem depends on the smoothness of the integrand $f$ . The more smooth $f$ is, the more accurately we can expect to compute $I[f]$ for a given budget of computational effort.

In this post, will measure smoothness using the reproducing kernel Hilbert space (RKHS) formalism. Let $\mathcal{H}$ be an RKHS with norm $\norm{\cdot}$ . We can interpret the norm as assigning a roughness $\norm{f}$ to each function $f$ . If $\norm{f}$ is large, then $f$ is rough; if $\norm{f}$ is small, then $f$ is smooth.

Associated to the RKHS $\mathcal{H}$ is the titular reproducing kernel $k$ . The kernel is a bivariate function $k:\Omega\times\Omega\to\real$ . It is related to the RKHS inner product $\langle\cdot,\cdot\rangle$ by the reproducing property

$f(x)=\langle f, k(x,\cdot)\rangle \quad \text{for every }f\in\mathcal{H},x\in\Omega.$

Here, $k(x,\cdot)$ represents the univariate function obtained by setting the first input of $k$ to be $x$ .

The Ideal Weights

To design a quadrature rule, we have to set the nodes $s = (s_1,\ldots,s_n) \in \Omega^n$ and weights $w = (w_1,\ldots,w_n)\in\real^n$ . Let’s first assume that the nodes $s$ are fixed, and talk about how to pick the weights $w$ .

There’s one choice of weights $w^\star$ that we’ll called the ideal weights. There (at least) are five equivalent ways of characterizing the ideal weights. We’ll present all of them. As an exercise, you can try and convince yourself that these characterizations are equivalent, giving rise to the same weights.

Interpretation 1: Exactness

A standard way of designing quadrature rules is to make them exact (i.e., error-free) for some class of functions. For instance, many classical quadrature rules are exact for polynomials of degree up to $n-1$ .

For kernel quadrature, it makes sense to design the quadrature rule to be exact for the kernel function at the selected nodes. That is, we require

$\hat{I}_{w_\star,s}[k(s_i,\cdot)]=I[k(s_i,\cdot)] \quad \text{for } i=1,2,\ldots,n.$

Enforcing exactness gives us $n$ linear equations for the $n$ unknowns $w^\star_1,\ldots,w^\star_n$ :

$\sum_{j=1}^n k(s_i,s_j)w^\star_j = \int_\Omega k(s_i,x) g(x)\,\mathrm{d}\mu(x) \quad \text{for }i=1,2,\ldots,n.$

Under mild conditions, this system of linear equations is uniquely solvable, and the solution $w^\star\in\real^n$ is the ideal weights.

Interpretation 2: Interpolate and Integrate

Here’s another very classical way of designing a quadrature rule. First, interpolate the function values $(s_i,f(s_i))$ at the nodes, obtaining an interpolant $\hat{f}$ . Then, obtain an approximation to the integral by integrating the interpolant:

$\hat{I}_{w^\star,s}[f] \coloneqq \int_\Omega \hat{f}(x) g(x) \, \mathrm{d}\mu(x).$

In our context, the appropriate interpolation method is kernel interpolation.¹ The kernel interpolant is defined to be the minimum-norm function $\hat{f}$ that interpolates the data:

$\hat{f} = \argmin \{ \norm{h} : h(s_i) = f(s_i) \text{ for } i=1,\ldots,n\}.$

Remarkably, this infinite-dimensional problem has a tractably computable solution. In fact, $\hat{f}$ is the unique function of the form

$\hat{f} = \sum_{i=1}^n \alpha_i k(\cdot,s_i)$

that agrees with $f$ on the points $s_1,\ldots,s_n$ .With a little algebra, you can show that the integral of $\hat{f}$ is

$I[\hat{f}] = \sum_{i=1}^n w^\star_i f(s_i),$

where $w^\star$ are the ideal weights.

Interpretation 3: Minimizing the Worst-Case Error

Define the worst-case error of weights $w$ and nodes $s$ to be

$\operatorname{Err}(w,s)=\sup_{\norm{f}\le 1}\left| I[f] - \hat{I}_{w,s}[f]\right|.$

The quantity $\operatorname{Err}(w,s)$ is the highest possible quadrature error for a function $f\in\mathcal{H}$ of norm at most 1.

Having defined the worst-case error, the ideal weights are precisely the weights that minimize this quantity

$w^\star=\operatorname*{argmin}_{w\in\real^n}\operatorname{Err}(w,s).$

Interpretation 4: Minimizing the Mean-Square Error

The next two interpretations of the ideal weights will adopt a probabilistic framing. A Gaussian process is a random function $f$ such that $f$ ’s values at any collection of points are (jointly) Gaussian random variables. We write $f\sim \operatorname{GP}(0,k)$ for a mean-zero Gaussian process with covariance function $k$ :

$\Cov(f(x),f(y))=k(x,y)\quad \text{for every } x,y\in\Omega.$

Define the mean-square quadrature error of weights $w$ and nodes $s$ to be

$\operatorname{MSE}(w,s)\coloneqq \expect_{f\sim\operatorname{GP}(0,k)} \left( I[f] - \hat{I}_{w,s}[f] \right)^2.$

The mean-square error reports the expected squared quadrature error over all functions $f$ drawn from a Gaussian process with covariance function $k$ .

Pleasantly, the mean-square error is equal ro the square of the worst-case error

$\operatorname{MSE}(w,s)=\operatorname{Err}(w,s)^2.$

As such, the ideal weights also minimize the mean-square error

$w^\star=\operatorname*{argmin}_{w\in\real^n}\operatorname{MSE}(w,s).$

Interpretation 5: Conditional Expectation

For our last interpretation, again consider a Gaussian process $h\sim \operatorname{GP}(0,k)$ . The integral of this random function $I[h]$ is a random variable. To numerically integrate a function $f$ , compute the expectation of $I[h]$ conditional on $h$ agreeing with $f$ at the quadrature nodes:

$\hat{I}_{w^\star,s}[f]\coloneqq \expect_{h\sim\operatorname{GP}(0,k)}[I[h] \mid h(s_i)=f(s_i) \text{ for } i=1,\ldots,n].$

One can show that this procedure yields the quadrature scheme with the ideal weights.

Conclusion

We’ve just seen five sensible ways of defining the ideal weights for quadrature in a general reproducing kernel Hilbert space. Remarkably, all five lead to exactly the same choice of weights. To me, these five equivalent characterizations give me more confidence that the ideal weights really are the “right” or “natural” choice for kernel quadrature.

That said, there are other reasonable requirements that we might want to impose on the weights. For instance, if $\mu$ is a probability measure and $g\equiv 1$ , it is reasonable to add an additional constraint that the weights $w$ lie in the probability simplex

$w\in\Delta\coloneqq \left\{ p\in\real^n_+ : \sum_{i=1}^n p_i = 1\right\}.$

With this additional stipulation, a quadrature rule can be interpreted as integrating $f$ against a discrete probability measure $\ \hat{\mu}=\sum_{i=1}^n w_i\delta_{s_i}$ ; thus, in effect, quadrature amounts to approximating one probability measure $\mu$ by another $\hat{\mu}$ . Additional constraints such as these can easily be imposed when using the optimization characterizations 3 and 4 of the ideal weights. See this paper for details.

What About the Nodes?

We’ve spent a lot of time talking about how to pick the quadrature weights, but how should we pick the nodes $s\in\Omega^n$ ? To pick the nodes, it seems sensible to try and minimize the worst-case error $\operatorname{Err}(w^\star,s)$ with the ideal weights $w^\star$ . For this purpose, we can use the following formula:

$\operatorname{Err}(w^\star,s) = \norm{\int_\Omega (k(\cdot,x) - \hat{k}_s(\cdot,x)) g(x) \, \mathrm{d}\mu(x)}.$

Here, $\hat{k}_s$ is the Nyström approximation to the kernel $k$ induced by the nodes $s$ , defined to be

$\hat{k}_s(x,y) = k(x,s) k(s,s)^{-1} k(s,y).$

We have written $k(s,s)$ for the kernel matrix with $ij$ entry $k(s_i,s_j)$ and $k(x,s)$ and $k(s,y)$ for the row and column vectors with $i$ th entry $k(x,s_i)$ and $k(s_i,y)$ .

I find the appearance of the Nyström approximation in this context to be surprising and delightful. Previously on this blog, we’ve seen (column) Nyström approximation in the context of matrix low-rank approximation. Now, a continuum analog of the matrix Nyström approximation has appeared in the error formula for numerical integration.

The appearance of the Nyström approximation in the kernel quadrature error $\operatorname{Err}(w^\star,s)$ also suggests a strategy for picking the nodes.

Node selection strategy. We should pick the nodes $s$ to make the Nyström approximation $\hat{k}_s \approx k$ as accurate as possible.

The closer $\hat{k}_s$ is to $k$ , the smaller the function $k(\cdot,x) - \hat{k}_s(\cdot,x)$ is and, thus, the smaller the error

$\operatorname{Err}(w^\star,s) = \norm{\int_\Omega (k(\cdot,x) - \hat{k}_s(\cdot,x)) g(x) \, \mathrm{d}\mu(x)}.$

Fortunately, we have randomized matrix algorithms for picking good nodes for matrix Nyström approximation such as randomly pivoted Cholesky, ridge leverage score sampling, and determinantal point process sampling; maybe these matrix tools can be ported to the continuous kernel setting?

Indeed, all three of these algorithms—randomly pivoted Cholesky, ridge leverage score sampling, and determinantal point process sampling—have been studied for kernel quadrature. The first of these algorithms, randomly pivoted Cholesky, is the subject of our paper. We show that this simple, adaptive sampling method produces excellent nodes for kernel quadrature. Intuitively, randomly pivoted Cholesky is effective because it is repulsive: After having picked nodes $s_1,\ldots,s_i$ , it has a high probability of placing the next node $s_{i+1}$ far from the previously selected nodes.

The following image shows 20 nodes selected by randomly pivoted Cholesky in a crescent-shaped region. The cyan–pink shading denotes the probability distribution for picking the next node. We see that the center of the crescent does not have any nodes, and thus is most likely to receive a node during the next round of sampling.

In our paper, we demonstrate—empirically and theoretically—that randomly pivoted Cholesky produces excellent nodes for quadrature. We also discuss efficient rejection sampling algorithms for sampling nodes with the randomly pivoted Cholesky distribution. Check out the paper for details!

Which Sketch Should I Use?

November 27, 2023 by Ethan N. Epperly 2 Comments

This is the second of a sequence of two posts on sketching, which I’m doing on the occasion of my new paper on the numerical stability of the iterative sketching method. For more on what sketching is and how it can be used to solve computational problems, I encourage you to check out the first post.

The goals of this post are more narrow. I seek to answer the question:

Which sketching matrix should I use?

To cut to the chase, my answer to this question is:

Sparse sign embeddings are a sensible default for sketching.

There are certainly cases when sparse sign embeddings are not the best type of sketch to use, but I hope to convince you that they’re a good sketching matrix to use for most purposes.

Experiments

Let’s start things off with some numerical experiments.¹ We’ll compare three types of sketching matrices: Gaussian embeddings, a subsampled randomized trigonometric transform (SRTT), and sparse sign embeddings. See the last post for descriptions of these sketching matrices. I’ll discuss a few additional types of sketching matrices that require more discussion at the end of this post.

Recall that a sketching matrix $S \in \real^{d\times n}$ seeks to compress a high-dimensional matrix $A \in \real^{n\times k}$ or vector $b\in\real^n$ to a lower-dimensional sketched matrix $SA$ or vector $Sb$ . The quality of a sketching matrix for a matrix $A$ is measured by its distortion $\varepsilon$ , defined to be the smallest number $\varepsilon > 0$ such that

$(1-\varepsilon) \norm{x} \le \norm{Sx} \le (1+\varepsilon) \norm{x} \quad \text{for every } x \in \operatorname{col}(A).$

Here, $\operatorname{col}(A)$ denotes the column space of the matrix $A$ .

Timing

We begin with timing test. We will measure three different times for each embedding:

Construction. The time required to generate the sketching matrix $S$ .
Vector apply. The time to apply the sketch to a single vector.
Matrix apply. The time to apply the sketch to an $n\times 200$ matrix.

We will test with input dimension $n = 10^6$ (one million) and output dimension $d = 400$ . For the SRTT, we use the discrete cosine transform as our trigonometric transform. For the sparse sign embedding, we use a sparsity parameter $\zeta = 8$ .

Here are the results (timings averaged over 20 trials):

Our conclusions are as follows:

Sparse sign embeddings are definitively the fastest to apply, being 3–20× faster than the SRTT and 74–100× faster than Gaussian embeddings.
Sparse sign embeddings are modestly slower to construct than the SRTT, but much faster to construct than Gaussian embeddings.

Overall, the conclusion is that sparse sign embeddings are the fastest sketching matrices by a wide margin: For an “end-to-end” workflow involving generating the sketching matrix $S \in \real^{400\times 10^6}$ and applying it to a matrix $A\in\real^{10^6\times 200}$ , sparse sign embeddings are 14× faster than SRTTs and 73× faster than Gaussian embeddings.²

Distortion

Runtime is only one measure of the quality of a sketching matrix; we also must care about the distortion $\varepsilon$ . Fortunately, for practical purposes, Gaussian embeddings, SRTTs, and sparse sign embeddings all tend to have similar distortions. Therefore, we are free to use the sparse sign embeddings, as they as typically are the fastest.

Consider the following test. We generate a sparse random test matrix $A$ of size $n\times k$ for $n = 10^5$ and $k = 50$ using the MATLAB sprand function; we set the sparsity level to 1%. We then compare the distortion of Gaussian embeddings, SRTTs, and sparse sign embeddings across a range of sketching dimensions $d$ between 100 and 10,000. We report the distortion averaged over 100 trials. The theoretically predicted value $\varepsilon = \sqrt{k/d}$ (equivalently, $d=k/\varepsilon^2$ ) is shown as a dashed line.

To me, I find these results remarkable. All three embeddings exhibit essentially the same distortion parameter predicted by the Gaussian theory.

It would be premature to declare success having only tested on one type of test matrix $A$ . Consider the following four test matrices:

Sparse: The test matrix from above.
Dense: $A\in\real^{10^6\times 50}$ is taken to be a matrix with independent standard Gaussian random values.
Khatri–Rao: $A\in\real^{50^3\times 50}$ is taken to be the Khatri–Rao product of three Haar random orthogonal matrices.
Identity: $A = \twobyone{I}{0} \in\real^{10^6\times 50}$ is taken to be the $50\times 50$ identity matrix stacked onto a $(10^6-50)\times 50$ matrix of zeros.

The performance of sparse sign embeddings (again with sparsity parameter $\zeta = 8$ ) is shown below:

We see that for the first three test matrices, the performance closely follows the expected value $\epsilon = \sqrt{k/d}$ . However, for the last test matrix “Identity”, we see the distortion begins to slightly exceed this predicted distortion for $d/k\ge 20$ .

To improve sparse sign embeddings for higher values of $d/k$ , we can increase the value of the sparsity parameter $\zeta$ . We recommend

$\zeta = \max \left( 8 , \left\lceil 2\sqrt{\frac{d}{k}} \right\rceil \right).$

With this higher sparsity level, the distortion closely tracks $\varepsilon = \sqrt{k/d}$ for all four test matrices:

Conclusion

Implemented appropriately (see below), sparse sign embeddings can be faster than other sketching matrices by a wide margin. The parameter choice $\zeta = 8$ is enough to ensure the distortion closely tracks $\varepsilon = \sqrt{k/d}$ for most test matrices. For the toughest test matrices, a slightly larger sparsity parameter $\zeta = \max(8, \lceil 2\sqrt{d/k}\rceil)$ can be necessary to achieve the optimal distortion.

While these tests are far from comprehensive, they are consistent with the uniformly positive results for sparse sign embeddings reported in the literature. We believe that this evidence supports the argument that sparse sign embeddings are a sensible default sketching matrix for most purposes.

Sparse Sign Embeddings: Theory and Practice

Given the highly appealing performance characteristics of sparse sign embeddings, it is worth saying a few more words about these embeddings and how they perform in both theory and practice.

Recall that a sparse sign embedding is a random matrix of the form

$S = \frac{1}{\sqrt{\zeta}} \begin{bmatrix} s_1 & \cdots & s_n \end{bmatrix}.$

Each column $s_i$ is an independent and randomly generated to contain exactly $\zeta$ nonzero entries in uniformly random positions. The value of each nonzero entry of $s_i$ is chosen to be either $+1$ or $-1$ with 50/50 odds.

Parameter Choices

The goal of sketching is to reduce vectors of length $n$ to a smaller dimension $d$ . For linear algebra applications, we typically want to preserve all vectors in the column space of a matrix $A$ up to distortion $\varepsilon > 0$ :

$(1-\varepsilon) \norm{x} \le \norm{Sx} \le (1+\varepsilon) \norm{x} \quad \text{for all }x \in \operatorname{col}(A).$

To use sparse sign embeddings, we must choose the parameters appropriately:

Given a dimension $k$ and a target distortion $\varepsilon$ , how do we pick $d$ and $\zeta$ ?

Based on the experiments above (and other testing reported in the literature), we recommend the following parameter choices in practice:

$d = \frac{k}{\varepsilon^2} \quad \text{and} \quad \zeta = \max\left(8,\frac{2}{\varepsilon}\right).$

The parameter choice $\zeta = 8$ is advocated by Tropp, Yurtever, Udell, and Cevher; they mention experimenting with parameter values as small as $\zeta = 2$ . The value $\zeta = 1$ has demonstrated deficiencies and should almost always be avoided (see below). The scaling $d \approx k/\varepsilon^2$ is derived from the analysis of Gaussian embeddings. As Martinsson and Tropp argue, the analysis of Gaussian embeddings tends to be reasonably descriptive of other well-designed random embeddings.

The best-known theoretical analysis, due to Cohen, suggests more cautious parameter setting for sparse sign embeddings:

$d = \mathcal{O} \left( \frac{k \log k}{\varepsilon^2} \right) \quad \text{and} \quad \zeta = \mathcal{O}\left( \frac{\log k}{\varepsilon} \right).$

The main difference between Cohen’s analysis and the parameter recommendations above is the presence of the $\log k$ factor and the lack of explicit constants in the O-notation.

Implementation

For good performance, it is imperative to store $S$ using either a purpose-built data structure or a sparse matrix format (such as a MATLAB sparse matrix or scipy sparse array).

If a sparse matrix library is unavailable, then either pursue a dedicated implementation or use a different type of embedding; sparse sign embeddings are just as slow as Gaussian embeddings if they are stored in an ordinary non-sparse matrix format!

Even with a sparse matrix format, it can require care to generate and populate the random entries of the matrix $S$ . Here, for instance, is a simple function for generating a sparse sign matrix in MATLAB:

function S = sparsesign_slow(d,n,zeta)
cols = kron((1:n)',ones(zeta,1)); % zeta nonzeros per column
vals = 2*randi(2,n*zeta,1) - 3; % uniform random +/-1 values
rows = zeros(n*zeta,1);
for i = 1:n
   rows((i-1)*zeta+1:i*zeta) = randsample(d,zeta);
end
S = sparse(rows, cols, vals / sqrt(zeta), d, n);
end

Here, we specify the rows, columns, and values of the nonzero entries before assembling them into a sparse matrix using the MATLAB sparse command. Since there are exactly $\zeta$ nonzeros per column, the column indices are easy to generate. The values are uniformly $\pm 1/\sqrt{\zeta}$ and can also be generated using a single line. The real challenge to generating sparse sign embeddings in MATLAB is the row indices, since each batch of $\zeta$ row indices much be chosen uniformly at random between $1$ and $d$ without replacement. This is accomplished in the above code by a for loop, generating row indices $\zeta$ at a time using the slow randsample function.

As its name suggests, the sparsesign_slow is very slow. To generate a $200\times 10^7$ sparse sign embedding with sparsity $\zeta = 8$ requires 53 seconds!

Fortunately, we can do (much) better. By rewriting the code in C and directly generating the sparse matrix in the CSC format MATLAB uses, generating this same 200 by 10 million sparse sign embedding takes just 0.4 seconds, a speedup of 130× over the slow MATLAB code. A C implementation of the sparse sign embedding that can be used in MATLAB using the MEX interface can be found in this file in the Github repo for my recent paper.

Other Sketching Matrices

Let’s leave off the discussion by mentioning other types of sketching matrices not considered in the empirical comparison above.

Coordinate Sampling

Another family of sketching matrices that we haven’t talked about are coordinate sampling sketches. A coordinate sampling sketch consists of indices $1\le i_1,\ldots,i_d\le n$ and weights $w_1,\ldots,w_d \in \real$ . To apply $S$ , we sample the indices $i_1,\ldots,i_d$ and reweight them using the weights:

$b \in \real^n \longmapsto Sb = (w_1 b_{i_1},\ldots,w_db_{i_d}) \in \real^d.$

Coordinate sampling is very appealing: To apply $S$ to a matrix or vector requires no matrix multiplication of trigonometric transforms, just picking out some entries or rows and rescaling them.

In order for coordinate sampling to be effective, we need to pick the right indices. Below, we compare two coordinate sampling sketching approaches, uniform sampling and leverage score sampling (both with replacement), to the sparse sign embedding with the suggested parameter setting $\zeta = \max(8,\lceil 2\sqrt{d/k}\rceil)$ for the hard “Identity” test matrix used above.

We see right away that the uniform sampling fails dramatically on this problem. That’s to be expected. All but 50 of 100,000 rows of $A$ are zero, so picking rows uniformly at random will give nonsense with very high probability. Uniform sampling can work well for matrices $A$ which are “incoherent”, with all rows of $A$ being of “similar importance”.

Conclusion (Uniform sampling). Uniform sampling is a risky method; it works excellently for some problems, but fails spectacularly for others. Use only with caution!

The ridge leverage score sampling method is more interesting. Unlike all the other sketches we’ve discussed in this post, ridge leverage score sampling is data-dependent. First, it computes a leverage score $\ell_i$ for each row of $A$ and then samples rows with probabilities proportional to these scores. For high enough values of $d$ , ridge leverage score sampling performs slightly (but only slightly) worse than the characteristic $\varepsilon = \sqrt{k/d}$ scaling we expect for an oblivious subspace embedding.

Ultimately, leverage score sampling has two disadvantages when compared with oblivious sketching matrices:

Higher distortion, higher variance. The distortion of a leverage score sketch is higher on average, and more variable, than an oblivious sketch, which achieve very consistent performance.
Computing the leverage scores. In order to implement this sketch, the leverage scores $\ell_i$ have to first be computed or estimated. This is a nontrivial algorithmic problem; the most direct way of computing the leverage scores requires a QR decomposition at $\mathcal{O}(nk^2)$ cost, much higher than other types of sketches.

There are settings when coordinate sampling methods, such as leverage scores, are well-justified:

Structured matrices. For some matrices $A$ , the leverage scores might be very cheap to compute or approximate. In such cases, coordinate sampling can be faster than oblivious sketching.
“Active learning”. For some problems, each entry of the vector $b$ or row of the matrix $A$ may be expensive to generate. In this case, coordinate sampling has the distinct advantage that computing $Sb$ or $SA$ only requires generating the entries of $b$ or rows of $A$ for the $d$ randomly selected indices $i_1,\ldots,i_d$ .

Ultimately, oblivious sketching and coordinate sampling both have their place as tools in the computational toolkit. For the reasons described above, I believe that oblivious sketching should usually be preferred to coordinate sampling in the absence of a special reason to prefer the latter.

Tensor Random Embeddings

There are a number of sketching matrices with tensor structure; see here for a survey. These types of sketching matrices are very well-suited to tensor computations. If tensor structure is present in your application, I would put these types of sketches at the top of my list for consideration.

CountSketch

The CountSketch sketching matrix is the $\zeta = 1$ case of the sparse sign embedding. CountSketch has serious deficiencies, and should only be used in practice with extreme care.

Consider the “Identity” test matrix from above but with parameter $k = 200$ , and compare the distortion of CountSketch to the sparse sign embedding with parameters $\zeta=2,4,8$ :

We see that the distortion of the CountSketch remains persistently high at 100% until the sketching dimension $d$ is taken $>4300$ , 20× higher than $k$ .

CountSketch is bad because it requires $d$ to be proportional to $k^2/\varepsilon^2$ in order to achieve distortion $\varepsilon$ . For all of the other sketching matrices we’ve considered, we’ve only required $d$ to be proportional to $k/\varepsilon^2$ (or perhaps $(k\log k)/\varepsilon^2$ ). This difference between $d\propto k^2$ for CountSketch and $d\propto k$ for other sketching matrices is a at the root of CountSketch’s woefully bad performance on some inputs.³

The fact that CountSketch requires $d\propto k^2$ is simple to show. It’s basically a variant on the famous birthday problem. We include a discussion at the end of this post.⁴

There are ways of fixing the CountSketch. For instance, we can use a composite sketch $S = S_2 \cdot S_1$ , where $S_1$ is a CountSketch of size $k^2/\varepsilon^2 \times n$ and $S_2$ is a Gaussian sketching matrix of size $k/\varepsilon^2 \times k^2/\varepsilon^2$ .⁵ For most applications, however, salvaging CountSketch doesn’t seem worth it; sparse sign embeddings with even $\zeta = 2$ nonzeros per column are already way more effective and reliable than a plain CountSketch.

Conclusion

By now, sketching is quite a big field, with dozens of different proposed constructions for sketching matrices. So which should you use? For most use cases, sparse sign embeddings are a good choice; they are fast to construct and apply and offer uniformly good distortion across a range of matrices.

Bonus: CountSketch and the Birthday Problem

The point of this bonus section is to prove the following (informal) theorem:

Let $A$ be the “Identity” test matrix above. If $S\in\real^{d\times n}$ is a CountSketch matrix with output dimension $d\ll k^2$ , then the distortion of $S$ for $\operatorname{col}(A)$ is $\varepsilon\ge 1$ with high probability.

Let’s see why. By the structure of the matrix $A$ , $SA$ has the form

$SA = \begin{bmatrix} s_1 & \cdots & s_k \end{bmatrix}$

where each vector $s_i\in\real^d$ has a single $\pm1$ in a uniformly random location $j\_i$ .

Suppose that the indices $j_1,\ldots,j_k$ are not all different from each other, say $j_i = j_{i'}$ . Set $x = e_i - e_{i'}$ , where $e_i$ is the standard basis vector with $1$ in position $i$ and zeros elsewhere. Then, $(SA)x = 0$ but $\norm{x} = \sqrt{2}$ . Thus, for the distortion relation

$(1-\varepsilon) \norm{x} =(1-\varepsilon)\sqrt{2} \le 0 = \norm{(SA)x}$

to hold, $\varepsilon \ge 1$ . Thus,

$\prob \{ \varepsilon \ge 1 \} \ge \prob \{ j_1,\ldots,j_k \text{ are not distinct} \}.$

For a moment, let’s put aside our analysis of the CountSketch, and turn our attention to a famous puzzle, known as the birthday problem:

How many people have to be in a room before there’s at least a 50% chance that two people share the same birthday?

The counterintuitive or “paradoxical” answer: 23. This is much smaller than many people’s intuition, as there are 365 possible birthdays and 23 is much smaller than 365.

The reason for this surprising result is that, in a room of 23 people, there are ${23 \choose 2} = 23\cdot 22/2=253$ pairs of people. Each pair of people has a $1/365$ chance of sharing a birthday, so the expected number of birthdays in a room of 23 people is $253/365 \approx 0.69$ . Since are 0.69 birthdays shared on average in a room of 23 people, it is perhaps less surprising that 23 is the critical number at which the chance of two people sharing a birthday exceeds 50%.

Hopefully, the similarity between the birthday problem and CountSketch is becoming clear. Each pair of indices $j_i$ and $j_{i'}$ in CountSketch have a $1/d$ chance of being the same. There are ${k\choose 2} \approx k^2/2$ pairs of indices, so the expected number of equal indices $j_i = j_{i'}$ is $\approx k^2/2d$ . Thus, we should anticipate $d \gtrapprox k^2$ is required to ensure that $j_1,\ldots,j_k$ are distinct with high probability.

Let’s calculate things out a bit more precisely. First, realize that

$\prob \{ j_1,\ldots,j_k \text{ are not distinct} \} = 1 - \prob \{ j_1,\ldots,j_k \text{ are distinct} \}.$

To compute the probability that $j_1,\ldots,j_k$ are distinct, imagine introducing each $j_i$ one at a time. Assuming that $j_1,\ldots,j_{i-1}$ are all distinct, the probability $j_1,\ldots,j_i$ are distinct is just the probability that $j_i$ does not take any of the $i-1$ values $j_1,\ldots,j_i$ . This probability is

$\prob\{ j_1,\ldots,j_i \text{ are distinct} \mid j_1,\ldots,j_{i-1} \text{ are distinct}\} = 1 - \frac{i-1}{d}.$

Thus, by the chain rule for probability,

$\prob \{ j_1,\ldots,j_k \text{ are distinct} \} = \prod_{i=1}^k \left(1 - \frac{i-1}{d} \right).$

To bound this quantity, use the numeric inequality $1-x\le \exp(-x)$ for every $x \in \real$ , obtaining

$\mathbb{P} \{ j_1,\ldots,j_k \text{ are distinct} \} \le \prod_{i=0}^{k-1} \exp\left(-\frac{i}{d}\right) = \exp \left( -\frac{1}{d}\sum_{i=0}^{k-1} i \right) = \exp\left(-\frac{k(k-1)}{2d}\right).$

Thus, we conclude that

$\prob \{ \varepsilon \ge 1 \} \ge 1-\prob \{ j_1,\ldots,j_k \text{ are distinct} \\}\ge 1-\exp\left(-\frac{k(k-1)}{2d}\right).$

Solving this inequality, we conclude that

$\prob\{\varepsilon \ge 1\} \ge \frac{1}{2} \quad \text{if} \quad d \le \frac{k(k-1)}{2\ln 2}.$

This is a quantitative version of our informal theorem from earlier.

Does Sketching Work?

November 13, 2023 by Ethan N. Epperly 4 Comments

I’m excited to share that my paper, Fast and forward stable randomized algorithms for linear least-squares problems has been released as a preprint on arXiv.

With the release of this paper, now seemed like a great time to discuss a topic I’ve been wanting to write about for a while: sketching. For the past two decades, sketching has become a widely used algorithmic tool in matrix computations. Despite this long history, questions still seem to be lingering about whether sketching really works:

does sketching actually work in practice? i want to know why it's not really used in deep learning
— typedfemale (@typedfemale) September 22, 2022

In this post, I want to take a critical look at the question “does sketching work”? Answering this question requires answering two basic questions:

What is sketching?
What would it mean for sketching to work?

I think a large part of the disagreement over the efficacy of sketching boils down to different answers to these questions. By considering different possible answers to these questions, I hope to provide a balanced perspective on the utility of sketching as an algorithmic primitive for solving linear algebra problems.

Sketching

In matrix computations, sketching is really a synonym for (linear) dimensionality reduction. Suppose we are solving a problem involving one or more high-dimensional vectors $b \in \real^n$ or perhaps a tall matrix $A\in \real^{n\times k}$ . A sketching matrix is a $d\times n$ matrix $S \in \real^{d\times n}$ where $d \ll n$ . When multiplied into a high-dimensional vector $b$ or tall matrix $A$ , the sketching matrix $S$ produces compressed or “sketched” versions $Sb$ and $SA$ that are much smaller than the original vector $b$ and matrix $A$ .

Let $\mathsf{E}=\{x_1,\ldots,x_p\}$ be a collection of vectors. For $S$ to be a “good” sketching matrix for $\mathsf{E}$ , we require that $S$ preserves the lengths of every vector in $\mathsf{E}$ up to a distortion parameter $\varepsilon>0$ :

(1) $(1-\varepsilon) \norm{x}\le\norm{Sx}\le(1+\varepsilon)\norm{x}$

for every $x$ in $\mathsf{E}$ .

For linear algebra problems, we often want to sketch a matrix $A$ . In this case, the appropriate set $\mathsf{E}$ that we want our sketch to be “good” for is the column space of the matrix $A$ , defined to be

$\operatorname{col}(A) \coloneqq \{ Ax : x \in \real^k \}.$

Remarkably, there exist many sketching matrices that achieve distortion $\varepsilon$ for $\mathsf{E}=\operatorname{col}(A)$ with an output dimension of roughly $d \approx k / \varepsilon^2$ . In particular, the sketching dimension $d$ is proportional to the number of columns $k$ of $A$ . This is pretty neat! We can design a single sketching matrix $S$ which preserves the lengths of all infinitely-many vectors $Ax$ in the column space of $A$ .

Sketching Matrices

There are many types of sketching matrices, each with different benefits and drawbacks. Many sketching matrices are based on randomized constructions in the sense that entries of $S$ are chosen to be random numbers. Broadly, sketching matrices can be classified into two types:

Data-dependent sketches. The sketching matrix $S$ is constructed to work for a specific set of input vectors $\mathsf{E}$ .
Oblivious sketches. The sketching matrix $S$ is designed to work for an arbitrary set of input vectors $\mathsf{E}$ of a given size (i.e., $\mathsf{E}$ has $p$ elements) or dimension ( $\mathsf{E}$ is a $k$ -dimensional linear subspace).

We will only discuss oblivious sketching for this post. We will look at three types of sketching matrices: Gaussian embeddings, subsampled randomized trignometric transforms, and sparse sign embeddings.

The details of how these sketching matrices are built and their strengths and weaknesses can be a little bit technical. All three constructions are independent from the rest of this article and can be skipped on a first reading. The main point is that good sketching matrices exist and are fast to apply: Reducing $b\in\real^n$ to $Sb\in\real^{d}$ requires roughly $\mathcal{O}(n\log n)$ operations, rather than the $\mathcal{O}(dn)$ operations we would expect to multiply a $d\times n$ matrix and a vector of length $n$ .¹

Gaussian Embeddings

The simplest type of sketching matrix $S\in\real^{d\times n}$ is obtained by (independently) setting every entry of $S$ to be a Gaussian random number with mean zero and variance $1/d$ . Such a sketching matrix is called a Gaussian embedding.²

Benefits. Gaussian embeddings are simple to code up, requiring only a standard matrix product to apply to a vector $Sb$ or matrix $SA$ . Gaussian embeddings admit a clean theoretical analysis, and their mathematical properties are well-understood.

Drawbacks. Computing $Sb$ for a Gaussian embedding costs $\mathcal{O}(dn)$ operations, significantly slower than the other sketching matrices we will consider below. Additionally, generating and storing a Gaussian embedding can be computationally expensive.

Subsampled Randomized Trigonometric Transforms

The subsampled randomized trigonometric transform (SRTT) sketching matrix takes a more complicated form. The sketching matrix is defined to be a scaled product of three matrices

$S = \sqrt{\frac{n}{d}} \cdot R \cdot F \cdot D.$

These matrices have the following definitions:

$D\in\real^{n\times n}$ is a diagonal matrix whose entries are each a random $\pm 1$ (chosen independently with equal probability).
$F\in\real^{n\times n}$ is a fast trigonometric transform such as a fast discrete cosine transform.³
$R\in\real^{d\times n}$ is a selection matrix. To generate $R$ , let $i_1,\ldots,i_d$ be a random subset of $\{1,\ldots,n\}$ , selected without replacement. $R$ is defined to be a matrix for which $Rb = (b_{i_1},\ldots,b_{i_d})$ for every vector $b$ .

To store $S$ on a computer, it is sufficient to store the $n$ diagonal entries of $D$ and the $d$ selected coordinates $i_1,\ldots,i_d$ defining $R$ . Multiplication of $S$ against a vector $b$ should be carried out by applying each of the matrices $R$ , $F$ , and $D$ in sequence, such as in the following MATLAB code:

% Generate randomness for S
signs = 2*randi(2,m,1)-3; % diagonal entries of D
idx = randsample(m,d); % indices i_1,...,i_d defining R

% Multiply S against b
c = signs .* b % multiply by D
c = dct(c) % multiply by F
c = c(idx) % multiply by R
c = sqrt(n/d) * c % scale

Benefits. $S$ can be applied to a vector $b$ in $\mathcal{O}(n \log n)$ operations, a significant improvement over the $\mathcal{O}(dn)$ cost of a Gaussian embedding. The SRTT has the lowest memory and random number generation requirements of any of the three sketches we discuss in this post.

Drawbacks. Applying $S$ to a vector requires a good implementation of a fast trigonometric transform. Even with a high-quality trig transform, SRTTs can be significantly slower than sparse sign embeddings (defined below).⁴ SRTTs are hard to parallelize.⁵ In theory, the sketching dimension should be chosen to be $d \approx (k\log k)/\varepsilon^2$ , larger than for a Gaussian sketch.

Sparse Sign Embeddings

A sparse sign embedding takes the form

$S = \frac{1}{\sqrt{\zeta}} \begin{bmatrix} s_1 & s_2 & \cdots & s_n \end{bmatrix}.$

Here, each column $s_i$ is an independently generated random vector with exactly $\zeta$ nonzero entries with random $\pm 1$ values in uniformly random positions. The result is a $d\times n$ matrix with only $\zeta \cdot n$ nonzero entries. The parameter $\zeta$ is often set to a small constant like $8$ in practice.⁶

Benefits. By using a dedicated sparse matrix library, $S$ can be very fast to apply to a vector $b$ (either $\mathcal{O}(n)$ or $\mathcal{O}(n\log k)$ operations) to apply to a vector, depending on parameter choices (see below). With a good sparse matrix library, sparse sign embeddings are often the fastest sketching matrix by a wide margin.

Drawbacks. To be fast, sparse sign embeddings requires a good sparse matrix library. They require generating and storing roughly $\zeta n$ random numbers, higher than SRTTs (roughly $n$ numbers) but much less than Gaussian embeddings ( $dn$ numbers). In theory, the sketching dimension should be chosen to be $d \approx (k\log k)/\varepsilon^2$ and the sparsity should be set to $\zeta \approx (\log k)/\varepsilon$ ; the theoretically sanctioned sketching dimension (at least according to existing theory) is larger than for a Gaussian sketch. In practice, we can often get away with using $d \approx k/\varepsilon^2$ and $\zeta=8$ .

Summary

Using either SRTTs or sparse maps, a sketching a vector of length $n$ down to $d$ dimensions requires only $\mathcal{O}(n)$ to $\mathcal{O}(n\log n)$ operations. To apply a sketch to an entire $n\times k$ matrix $A$ thus requires roughly $\mathcal{O}(nk)$ operations. Therefore, sketching offers the promise of speeding up linear algebraic computations involving $A$ , which typically take $\mathcal{O}(nk^2)$ operations.

How Can You Use Sketching?

The simplest way to use sketching is to first apply the sketch to dimensionality-reduce all of your data and then apply a standard algorithm to solve the problem using the reduced data. This approach to using sketching is called sketch-and-solve.

As an example, let’s apply sketch-and-solve to the least-squares problem:

(2) $\operatorname*{minimize}_{x\in\real^k} \norm{Ax - b}.$

We assume this problem is highly overdetermined with $A$ having many more rows $n$ than columns $m$ .

To solve this problem with sketch-and-solve, generate a good sketching matrix $S$ for the set $\mathsf{E} = \operatorname{col}(\onebytwo{A}{b})$ . Applying $S$ to our data $A$ and $b$ , we get a dimensionality-reduced least-squares problem

(3) $\operatorname*{minimize}_{\hat{x}\in\real^k} \norm{(SA)\hat{x} - Sb}.$

The solution $\hat{x}$ is the sketch-and-solve solution to the least-squares problem, which we can use as an approximate solution to the original least-squares problem.

Least-squares is just one example of the sketch-and-solve paradigm. We can also use sketching to accelerate other algorithms. For instance, we could apply sketch-and-solve to clustering. To cluster data points $x_1,\ldots,x_p$ , first apply sketching to obtain $Sx_1,\ldots,Sx_p$ and then apply an out-of-the-box clustering algorithms like k-means to the sketched data points.

Does Sketching Work?

Most often, when sketching critics say “sketching doesn’t work”, what they mean is “sketch-and-solve doesn’t work”.

To address this question in a more concrete setting, let’s go back to the least-squares problem (2). Let $x_\star$ denote the optimal least-squares solution and let $\hat{x}$ be the sketch-and-solve solution (3). Then, using the distortion condition (1), one can show that

$\norm{A\hat{x} - b} \le \frac{1+\varepsilon}{1-\varepsilon} \norm{Ax - b}.$

If we use a sketching matrix with a distortion of $\varepsilon = 1/3$ , then this bound tells us that

(4) $\norm{A\hat{x} - b} \le 2\norm{Ax_\star - b}.$

Is this a good result or a bad result? Ultimately, it depends. In some applications, the quality of a putative least-squares solution $\hat{x}$ is can be assessed from the residual norm $\norm{A\hat{x} - b}$ . For such applications, the bound (4) ensures that $\norm{A\hat{x} - b}$ is at most twice $\norm{Ax_\star-b}$ . Often, this means $\hat{x}$ is a pretty decent approximate solution to the least-squares problem.

For other problems, the appropriate measure of accuracy is the so-called forward error $\norm{\hat{x} - x_\star}$ , measuring how close $\hat{x}$ is to $x_\star$ . For these cases, it is possible that $\norm{\hat{x} - x_\star}$ might be large even though the residuals are comparable (4).

Let’s see an example, using the MATLAB code from my paper:

[A, b, x, r] = random_ls_problem(1e4, 1e2, 1e8, 1e-4); % Random LS problem
S = sparsesign(4e2, 1e4, 8); % Sparse sign embedding
sketch_and_solve = (S*A) \ (S*b); % Sketch-and-solve
direct = A \ b; % MATLAB mldivide

Here, we generate a random least-squares problem of size 10,000 by 100 (with condition number $10^8$ and residual norm $10^{-4}$ ). Then, we generate a sparse sign embedding of dimension $d = 400$ (corresponding to a distortion of roughly $\varepsilon \approx 1/2$ ). Then, we compute the sketch-and-solve solution and, as reference, a “direct” solution by MATLAB’s \.

We compare the quality of the sketch-and-solve solution to the direct solution, using both the residual and forward error:

fprintf('Residuals: sketch-and-solve %.2e, direct %.2e, optimal %.2e\n',...
           norm(b-A*sketch_and_solve), norm(b-A*direct), norm(r))
fprintf('Forward errors: sketch-and-solve %.2e, direct %.2e\n',...
           norm(x-sketch_and_solve), norm(x-direct))

Here’s the output:

Residuals: sketch-and-solve 1.13e-04, direct 1.00e-04, optimal 1.00e-04
Forward errors: sketch-and-solve 1.06e+03, direct 8.08e-07

The sketch-and-solve solution has a residual norm of $1.13\times 10^{-4}$ , close to direct method’s residual norm of $1.00\times 10^{-4}$ . However, the forward error of sketch-and-solve is $1\times 10^3$ nine orders of magnitude larger than the direct method’s forward error of $8\times 10^{-7}$ .

Does sketch-and-solve work? Ultimately, it’s a question of what kind of accuracy you need for your application. If a small-enough residual is all that’s needed, then sketch-and-solve is perfectly adequate. If small forward error is needed, sketch-and-solve can be quite bad.

One way sketch-and-solve can be improved is by increasing the sketching dimension $d$ and lowering the distortion $\varepsilon$ . Unfortunately, improving the distortion of the sketch is expensive. Because of the relation $d \approx k /\varepsilon^2$ , to decrease the distortion by a factor of ten requires increasing the sketching dimension $d$ by a factor of one hundred! Thus, sketch-and-solve is really only appropriate when a low degree of distortion $\varepsilon$ is necessary.

Iterative Sketching: Combining Sketching with Iteration

Sketch-and-solve is a fast way to get a low-accuracy solution to a least-squares problem. But it’s not the only way to use sketching for least-squares. One can also use sketching to obtain high-accuracy solutions by combining sketching with an iterative method.

There are many iterative methods for least-square problems. Iterative methods generate a sequence of approximate solutions $x_1,x_2,\ldots$ that we hope will converge at a rapid rate to the true least-squares solution, $x_\star$ .

To using sketching to solve least-squares problems iteratively, we can use the following observation:

If $S$ is a sketching matrix for $\mathsf{E} = \operatorname{col}(A)$ , then $(SA)^\top SA \approx A^\top A$ .

Therefore, if we compute a QR factorization

$SA = QR,$

then

$A^\top A \approx (SA)^\top (SA) = R^\top Q^\top Q R = R^\top R.$

Notice that we used the fact that $Q^\top Q = I$ since $Q$ has orthonormal columns. The conclusion is that $R^\top R \approx A^\top A$ .

Let’s use the approximation $R^\top R \approx A^\top A$ to solve the least-squares problem iteratively. Start off with the normal equations⁷

(5) $(A^\top A)x = A^\top b.$

We can obtain an approximate solution to the least-squares problem by replacing $A^\top A$ by $R^\top R$ in (5) and solving. The resulting solution is

$x_0 = R^{-1} (R^{-\top}(A^\top b)).$

This solution $x_0$ will typically not be a good solution to the least-squares problem (2), so we need to iterate. To do so, we’ll try and solve for the error $x - x_0$ . To derive an equation for the error, subtract $A^\top A x_0$ from both sides of the normal equations (5), yielding

$(A^\top A)(x-x_0) = A^\top (b-Ax_0).$

Now, to solve for the error, substitute $R^\top R$ for $A^\top A$ again and solve for $x$ , obtaining a new approximate solution $x_1$ :

$x\approx x_1 \coloneqq x_0 + R^{-\top}(R^{-1}(A^\top(b-Ax_0))).$

We can now go another step: Derive an equation for the error $x-x_1$ , approximate $A^\top A \approx R^\top R$ , and obtain a new approximate solution $x_2$ . Continuing this process, we obtain an iteration

(6) $x_{i+1} = x_i + R^{-\top}(R^{-1}(A^\top(b-Ax_i))).$

This iteration is known as the iterative sketching method.⁸ If the distortion is small enough, this method converges at an exponential rate, yielding a high-accuracy least squares solution after a few iterations.

Let’s apply iterative sketching to the example we considered above. We show the forward error of the sketch-and-solve and direct methods as horizontal dashed purple and red lines. Iterative sketching begins at roughly the forward error of sketch-and-solve, with the error decreasing at an exponential rate until it reaches that of the direct method over the course of fourteen iterations. For this problem, at least, iterative sketching gives high-accuracy solutions to the least-squares problem!

To summarize, we’ve now seen two very different ways of using sketching:

Sketch-and-solve. Sketch the data $A$ and $b$ and solve the sketched least-squares problem (3). The resulting solution $\hat{x}$ is cheap to obtain, but may have low accuracy.
Iterative sketching. Sketch the matrix $A$ and obtain an approximation $R^\top R = (SA)^\top (SA)$ to $A^\top A$ . Use the approximation $R^\top R$ to produce a sequence of better-and-better least-squares solutions $x_i$ by the iteration (6). If we run for enough iterations $q$ , the accuracy of the iterative sketching solution $x_q$ can be quite high.

By combining sketching with more sophisticated iterative methods such as conjugate gradient and LSQR, we can get an even faster-converging least-squares algorithm, known as sketch-and-precondition. Here’s the same plot from above with sketch-and-precondition added; we see that sketch-and-precondition converges even faster than iterative sketching does!

“Does sketching work?” Even for a simple problem like least-squares, the answer is complicated:

A direct use of sketching (i.e., sketch-and-solve) leads to a fast, low-accuracy solution to least-squares problems. But sketching can achieve much higher accuracy for least-squares problems by combining sketching with an iterative method (iterative sketching and sketch-and-precondition).

We’ve focused on least-squares problems in this section, but these conclusions could hold more generally. If “sketching doesn’t work” in your application, maybe it would if it was combined with an iterative method.

Just How Accurate Can Sketching Be?

We left our discussion of sketching-plus-iterative-methods in the previous section on a positive note, but there is one last lingering question that remains to be answered. We stated that iterative sketching (and sketch-and-precondition) converge at an exponential rate. But our computers store numbers to only so much precision; in practice, the accuracy of an iterative method has to saturate at some point.

An (iterative) least-squares solver is said to be forward stable if, when run for a sufficient number $q$ of iterations, the final accuracy $\norm{x_q - x_\star}$ is comparable to accuracy of a standard direct method for the least-squares problem like MATLAB’s \ command or Python’s scipy.linalg.lstsq. Forward stability is not about speed or rate of convergence but about the maximum achievable accuracy.

The stability of sketch-and-precondition was studied in a recent paper by Meier, Nakatsukasa, Townsend, and Webb. They demonstrated that, with the initial iterate $x_0 = 0$ , sketch-and-precondition is not forward stable. The maximum achievable accuracy was worse than standard solvers by orders of magnitude! Maybe sketching doesn’t work after all?

Fortunately, there is good news:

The iterative sketching method is provably forward stable. This result is shown in my newly released paper; check it out if you’re interested!
If we use the sketch-and-solve method as the initial iterate $x_0 = \hat{x}$ for sketch-and-precondition, then sketch-and-precondition appears to be forward stable in practice. No theoretical analysis supporting this finding is known at present.⁹

These conclusions are pretty nuanced. To see what’s going, it can be helpful to look at a graph:¹⁰

The performance of different methods can be summarized as follows: Sketch-and-solve can have very poor forward error. Sketch-and-precondition with the zero initialization $x_0 = 0$ is better, but still much worse than the direct method. Iterative sketching and sketch-and-precondition with $x_0 = \hat{x}$ fair much better, eventually achieving an accuracy comparable to the direct method.

Put more simply, appropriately implemented, sketching works after all!

Conclusion

Sketching is a computational tool, just like the fast Fourier transform or the randomized SVD. Sketching can be used effectively to solve some problems. But, like any computational tool, sketching is not a silver bullet. Sketching allows you to dimensionality-reduce matrices and vectors, but it distorts them by an appreciable amount. Whether or not this distortion is something you can live with depends on your problem (how much accuracy do you need?) and how you use the sketch (sketch-and-solve or with an iterative method).

The Hard Way to Prove Jensen’s Inequality

June 5, 2023 by Ethan N. Epperly Leave a comment

In this post, I want to discuss a very beautiful piece of mathematics I stumbled upon recently. As a warning, this post will be more mathematical than most, but I will still try and sand off the roughest mathematical edges. This post is adapted from a much more comprehensive post by Paata Ivanishvili. My goal is to distill the main idea to its essence, deferring the stochastic calculus until it cannot be avoided.

Jensen’s inequality is one of the most important results in probability.

Jensen’s inequality. Let $X$ be a (real) random variable and $f:\real\to\real$ a convex function such that both $\mathbb{E} X$ and $\mathbb{E} f(X)$ are defined. Then $f(\mathbb{E}X) \le \mathbb{E} [f(X)]$ .

Here is the standard proof. A convex function has supporting lines. That is, at a point $a \in \real$ , there exists a slope $m$ such that $m(x-a) + f(a) \le f(x)$ for all $x \in \real$ . Invoke this result at $a = \mathbb{E} X$ and $x = X$ and take expectations to conclude that

$\mathbb{E}[m(X - \mathbb{E}X) + f(\mathbb{E}X)] = f(\mathbb{E}X) \le \mathbb{E} [f(X)].$

In this post, I will outline a proof of Jensen’s inequality which is much longer and more complicated. Why do this? This more difficult proof illustrates an incredible powerful technique for proving inequalities, interpolation. The interpolation method can be used to prove a number of difficult and useful inequalities in probability theory and beyond. As an example, at the end of this post, we will see the Gaussian Jensen inequality, a striking generalization of Jensen’s inequality with many applications.

The idea of interpolation is as follows: Suppose I wish to prove $A_0 \le A_1$ for two numbers $A_0$ and $A_1$ . This may hard to do directly. With the interpolation method, I first construct a family of numbers $A_t$ , $0 \le t \le 1$ , such that $A_{t = 0} = A_0$ and $A_{t=1} = A_1$ and show that $(A_t : 0\le t\le 1)$ is (weakly) increasing in $t$ . This is typically accomplished by showing the derivative is nonnegative:

$\frac{d}{dt} A_t \ge 0.$

To prove Jensen’s inequality by interpolation, we shall begin with a special case. As often in probability, the simplest case is that of a Gaussian random variable.

Jensen’s inequality for a Gaussian. Let $X$ be a standard Gaussian random variable (i.e., mean-zero and variance $1$ ) and let $f:\real\to\real$ be a thrice-differentiable convex function satisfying a certain technical condition.¹ Then
$f(0) \le \mathbb{E} [f(X)].$

Note that the conclusion is exactly Jensen’s inequality, as we have assumed $X$ is mean-zero.

The difficulty with any proof by interpolation is to come up with the “right” $A_t$ . For us, the “right” answer will take the form

$A_t = \mathbb{E} [ f(X_t) ],$

where $X_0 = 0$ starts with no randomness and $X_1 = X$ is our standard Gaussian. To interpolate between these extremes, we increase the variance linearly from $0$ to $1$ . Thus, we define

$A_t = \mathbb{E} [ f(X_t)] \quad \text{where $X_t\sim\mathcal{N}(0,t)$}.$

Here, and throughout, $\mathcal{N}(0,v)$ denotes a Gaussian random variable with zero mean and variance $v$ .

Let’s compute the derivative of $A_t$ . To do this, let $\delta > 0$ denote a small parameter which we will later send to zero. For us, the key fact will be that a $\mathcal{N}(0,t+\delta)$ can be realized as a sum of independent $\mathcal{N}(0,t)$ and $\mathcal{N}(0,\delta)$ random variables. Therefore, we write

$X_{t+\delta} = X_t + \Delta \quad \text{where $\Delta \sim \mathcal{N}(0,\delta)$ is independent of $X_t$.}$

We now evaluate $f(X_t+\Delta)$ by using Taylor’s formula

(1) $f(X_t+\Delta) = f(X_t) + f'(X_t)\Delta + \frac{1}{2} f''(X_t) \Delta^2 + \frac{1}{6} f'''(\xi) \Delta^3,$

where $\xi$ lies between $X_t$ and $X_t+\Delta$ . Now, take expectations,

$\mathbb{E}[ f(X_t+\Delta)]=\mathbb{E}[f(X_t)] + \mathbb{E}[f'(X_t)\Delta] + \frac{1}{2} \mathbb{E}[f''(X_t)] \mathbb{E}[\Delta^2] + \underbrace{\frac{1}{6} \mathbb{E}[f'''(\xi) \Delta^3]}_{:=\mathrm{Rem}(\delta)}.$

The random variable $\Delta$ has mean zero and variance $\delta$ so this gives

$\mathbb{E} [f(X_t+\Delta)]=\mathbb{E}[f(X_t)] + \delta \frac{1}{2} \mathbb{E}[f''(X_t)] + \mathrm{Rem}(\delta).$

As we show below, the remainder term $\mathrm{Rem}(\delta)/\delta$ vanishes as $\delta\to 0$ . Thus, we can rearrange this expression to compute the derivative:

$\frac{d}{dt} A_t = \lim_{\delta \downarrow 0} \frac{\mathbb{E} f(X_t+\Delta)-\mathbb{E}[f(X_t)]}{\delta} = \lim_{\delta \downarrow 0} \frac{1}{2} \mathbb{E}[f''(X_t)] + \frac{\mathrm{Rem}(\delta)}{\delta} = \frac{1}{2} \mathbb{E}[f''(X_t)].$

The second derivative of a convex function is nonnegative: $f''(x) \ge 0$ for every $x$ . Therefore,

$\frac{d}{dt} A_t \ge 0 \quad \text{for all } t\in [0,1].$

Jensen’s inequality is proven! In fact, we’ve proven the stronger version of Jensen’s inequality:

$\mathbb{E} f(X) = f(0) + \frac{1}{2} \int_0^1 \mathbb{E} [f''(X_t)] \, dt.$

This strengthened version can yield improvements. For instance, if $f$ is $\beta$ -smooth

$f''(x) \le \beta \quad \text{for every } x \in \real,$

then we have

$f(0) \le \mathbb{E} f(X) \le f(0) + \frac{1}{2}\beta.$

This inequality isn’t too hard to prove directly, but it does show that we’ve obtained something more than the simple proof of Jensen’s inequality.

Analyzing the Remainder Term

Let us quickly check that the remainder term vanishes $\mathrm{Rem}(\delta)/\delta$ as $\delta \to 0$ . Let’s do this. As an exercise, you can verify that our technical regularity condition implies $\mathbb{E} |f'''(\xi)|^p < +\infty$ . Thus, by Hölder’s inequality and setting $q$ to be $p$ ‘s Hölder conjugate ( $1/p = 1/q = 1$ ), we obtain

$\frac{|\mathrm{Rem}(\delta)|}{\delta} = \frac{|\mathbb{E}[f'''(\xi) \Delta^3]|}{6\delta} \le \frac{(|\mathbb{E} |f'''(\xi)|^p)^{1/p}| (\mathbb{E} |\Delta|^{3q})^{1/q}}{6\delta}.$

One can show that $(\mathbb{E} |\Delta|^{3q})^{1/q} \le C(q) \delta^{3/2}$ where $C(q)$ is a function of $q$ alone. Therefore, $|\mathrm{Rem}(\delta)|/\delta \le \mathrm{constant} \cdot \delta^{1/2} \to 0$ as $\delta \downarrow 0$ .

What’s Really Going On Here?

In our proof, we use a family of random variables $X_t \sim \mathcal{N}(0,t)$ , defined for each $0\le t \le 1$ . Rather than treating these quantities as independent, we can think of them as a collective, comprising a random function $t \mapsto X_t$ known as a Brownian motion.

The Brownian motion is a very natural way of interpolating between a constant $\mu$ and a Gaussian with mean $\mu$ .²

There is an entire subject known as stochastic calculus which allows us to perform computations with Brownian motion and other random processes. The rules of stochastic calculus can seem bizarre at first. For a function $f$ of a real number $x$ , we often write

$df = f'(x) \, dx$

For a function $f(X_t)$ of a Brownian motion, the analog is Itô’s formula

$df = f'(X_t) \, dX_t + \frac{1}{2} f''(X_t) \, dt.$

While this might seem odd at first, this formula may seem more sensible if we compare with (1) above. The idea, very roughly, is that for an increment of the Brownian motion $dX_t$ over a time interval $dt$ , $(dX_t)^2$ is a random variable with mean $dt$ , so we cannot drop the second term in the Taylor series, even up to first order in $dt$ . Fully diving into the subtleties of stochastic calculus is far beyond the scope of this short post. Hopefully, the rest of this post, which outlines some extensions of our proof of Jensen’s inequality that require more stochastic calculus, will serve as an enticement to learn more about this beautiful subject.

Proving Jensen by Interpolation

For the rest of this post, we will be less careful with mathematical technicalities. We can use the same idea that we used to prove Jensen’s inequality for a Gaussian random variable to prove Jensen’s inequality for any random variable $Y$ :

$f(\mathbb{E}Y) \le \mathbb{E}[f(Y)].$

Here is the idea of the proof.

First, realize that we can write any random variable $Y$ as a function of a standard Gaussian random variable $X$ . Indeed, letting $F_X$ and $F_Y$ denote the cumulative distribution functions of $X$ and $Y$ , one can show that

$g(X) := \inf \{ \alpha \in \real : F_Y(\alpha) \ge F_X(X) \}$

has the same distribution as $Y$ .

Now, as before, we can interpolate between $\mathbb{E} Y$ and $Y$ using a Brownian motion. As a first, idea, we might try

$A_t \stackrel{?}{=} \mathbb{E} [f(g(X_t))].$

Unfortunately, this choice of $A_t$ does not work! Indeed, $A_0 = \mathbb{E}[f(g(0))]$ does not even equal to $\mathbb{E} [f(Y)]$ ! Instead, we must define

$A_t = \mathbb{E} [f(\mathbb{E}[g(X_1) \mid X_t])].$

We define $A_t$ using the conditional expectation of the final value $g(X_1)$ conditional on the Brownian motion $X_t$ at an earlier time $t$ . Using a bit of elbow grease and stochastic calculus, one can show that

$\frac{d}{dt} A_t \ge 0 \quad \text{for all }t\in [0,1].$

This provides a proof of Jensen’s inequality in general by the method if interpolation.

Gaussian Jensen Inequality

Now, we’ve come to the real treat, the Gaussian Jensen inequality. In the last section, we saw the sketch of a proof of Jensen’s inequality using interpolation. While it is cool that this proof is possible, we learned anything new since we can prove Jensen’s inequality in other ways. The Gaussian Jensen inequality provides an application of this technique which is hard to prove other ways. This section, in particular, is cribbing quite heavily from Paata Ivanishvili‘s excellent post on the topic.

Here’s the big question:

If $Y_1,\ldots,Y_n$ are “somewhat dependent”, for which functions does the multivariate Jensen’s inequality
( $\star$ ) $f(\mathbb{E} Y_1,\ldots,\mathbb{E}Y_n) \le \mathbb{E} [f(Y_1,\ldots,Y_n)]$
hold?

Considering extreme cases, if $Y_1,\ldots,Y_n$ are entirely dependent, then we would only expect ( $\star$ ) to hold when $f$ is convex. But if $Y_1,\ldots,Y_n$ are independent, then we can apply Jensen’s inequality to each coordinate one at a time to deduce

$\text{($\star$) holds if $f$ is convex in each coordinate, separately.}$

We would like a result which interpolates between extremes {fully dependent, fully convex} and {independent, separately convex}. The Gaussian Jensen inequality provides exactly this tool.

As in the previous section, we can generate arbitrary random variables $Y_1,\ldots,Y_n$ as functions $g(X_1),\ldots,g(X_n)$ of Gaussian random variables $X_1,\ldots,X_n$ . We will use the covariance matrix $\Sigma$ of the Gaussian random variables $X_1,\ldots,X_n$ as our measure of the dependence of the random variables $Y_1,\ldots,Y_n$ . With this preparation in place, we have the following result:

Gaussian Jensen inequality. The conclusion of Jensen’s inequality
(2) $f(\mathbb{E}g_1(X_1),\ldots,\mathbb{E}g_n(X_n)) \le \mathbb{E} [f(g(X_1),\ldots,g(X_n))]$
holds for all test functions $g_1,\ldots,g_n$ if and only if
$\Sigma \circ \nabla^2 f(x) \text{ is positive semidefinite} \quad \text{for all $x \in \real^n$}.$
Here, $\nabla^2 f(x)$ is the Hessian matrix at $x$ and $\circ$ denotes the entrywise product of matrices.

This is a beautiful result with striking consequences (see Ivanishvili‘s post). The proof is essentially the same as the proof as Jensen’s inequality by interpolation with a little additional bookkeeping.

Let us confirm this result respects our extreme cases. In the case where $X_1=\cdots=X_n$ are equal (and variance one), $\Sigma$ is a matrix of all ones and $\Sigma \circ \nabla^2 f(x) = \nabla^2 f(x)$ for all $x$ . Thus, the Gaussian Jensen inequality states that (2) holds if and only if $\nabla^2 f(x)$ is positive semidefinite for every $x$ , which occurs precisely when $f$ is convex.

Next, suppose that $X_1,\ldots,X_n$ are independent and variance one, then $\Sigma$ is the identity matrix and

$\Sigma \circ \nabla^2 f(x) = \mathrm{diag} \left( \frac{\partial^2 f}{\partial x_i^2} : i=1,\ldots,n \right).$

A diagonal matrix is positive semidefinite if and only if its entries are nonnegative. Thus, (2) holds if and only if each of $f$ ‘s diagonal second derivatives are nonnegative $\partial_{x_i}^2 f \ge 0$ : this is precisely the condition for $f$ to be separately convex in each argument.

There’s much more to be said about the Gaussian Jensen inequality, and I encourage you to read Ivanishvili‘s post to see the proof and applications. What I find so compelling about this result—so compelling that I felt the need to write this post—is how interpolation and stochastic calculus can be used to prove inequalities which don’t feel like stochastic calculus problems. The Gaussian Jensen inequality is a statement about functions of dependent Gaussian random variables; there’s nothing dynamic happening. Yet, to prove this result, we inject dynamics into the problem, viewing the two sides of our inequality as endpoints of a random process connecting them. This is a such a beautiful idea that I couldn’t help but share it.

Distribution	Variance (divided by $\tr(A)^2$ )
Gaussian	$2.0\times 10^{-3}$
Sphere	$6.7\times 10^{-6}$
Random signs	$6.7\times 10^{-6}$

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Motivating Application: Randomized Kaczmarz

Rejection Sampling in General

Application: Randomly pivoted Cholesky

Conclusion: Rejection Sampling, an Algorithm Design Tool

Which Distribution Should You Use: Signs vs. Sphere

The MMMW Trace Estimation “Speed Limit”

Communication Complexity

Reducing Gap-Hamming to Trace Estimation

Protocol

Analysis

Integration and Quadrature

Smoothness and Reproducing Kernel Hilbert Spaces

The Ideal Weights

Interpretation 1: Exactness

Interpretation 2: Interpolate and Integrate

Interpretation 3: Minimizing the Worst-Case Error

Interpretation 4: Minimizing the Mean-Square Error

Interpretation 5: Conditional Expectation

Conclusion

What About the Nodes?

Experiments

Timing

Distortion

Conclusion

Sparse Sign Embeddings: Theory and Practice

Parameter Choices

Implementation

Other Sketching Matrices

Coordinate Sampling

Tensor Random Embeddings

CountSketch

Conclusion

Sketching

Sketching Matrices

Summary

How Can You Use Sketching?

Does Sketching Work?

Iterative Sketching: Combining Sketching with Iteration

Just How Accurate Can Sketching Be?

Conclusion

What’s Really Going On Here?

Proving Jensen by Interpolation

Gaussian Jensen Inequality