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We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an n × n matrix A, accessible only though matrix-vector products with A and AT. We prove that, for the rank-k HODLR approximation problem, our method achieves a (1 + β )log(n )-optimal approximation in expected Frobenius norm using O (k log(n )/β3) matrix-vector products. In particular, the algorithm obtains a (1 + ∈ )-optimal approximation with O (k log4(n )/∈3) matrix-vector products, and for any constant c, an nc-optimal approximation with O (k log(n )) matrix-vector products. Apart from matrix-vector products, the additional computational cost of our method is just O (n poly(log(n ), k, β )). We complement the upper bound with a lower bound, which shows that any matrix-vector query algorithm requires at least Ω(k log(n ) + k/ε ) queries to obtain a (1 + ε )-optimal approximation. Our algorithm can be viewed as a robust version of widely used “peeling” methods for recovering HODLR matrices and is, to the best of our knowledge, the first matrix-vector query algorithm to enjoy theoretical worst- case guarantees for approximation by any hierarchical matrix class. To control the propagation of error between levels of hierarchical approximation, we introduce a new perturbation bound for low-rank approximation, which shows that the widely used Generalized Nyström method enjoys inherent stability when implemented with noisy matrix-vector products. We also introduce a novel randomly perforated matrix sketching method to further control the error in the peeling algorithm. 

We study algorithms for approximating the spectral density (i.e., the eigenvalue distribution) of a symmetric matrix A ∈ ℝn×n that is accessed through matrix-vector product queries. Recent work has analyzed popular Krylov subspace methods for this problem, showing that they output an ∈ · || A||2 error approximation to the spectral density in the Wasserstein-1 metric using O (1/∈ ) matrix-vector products. By combining a previously studied Chebyshev polynomial moment matching method with a deflation step that approximately projects off the largest magnitude eigendirections of A before estimating the spectral density, we give an improved error bound of ∈ · σℓ (A) using O (ℓ log n + 1/∈ ) matrix-vector products, where σℓ (A) is the ℓth largest singular value of A. In the common case when A exhibits fast singular value decay and so σℓ (A) « ||A||2, our bound can be much stronger than prior work. We also show that it is nearly tight: any algorithm giving error ∈ · σℓ (A) must use Ω(ℓ + 1/∈ ) matrix-vector products. We further show that the popular Stochastic Lanczos Quadrature (SLQ) method essentially matches the above bound for any choice of parameter ℓ, even though SLQ itself is parameter-free and performs no explicit deflation. Our bound helps to explain the strong practical performance and observed ‘spectrum adaptive’ nature of SLQ, and motivates a simple variant of the method that achieves an even tighter error bound. Technically, our results require a careful analysis of how eigenvalues and eigenvectors are approximated by (block) Krylov subspace methods, which may be of independent interest. Our error bound for SLQ leverages an analysis of the method that views it as an implicit polynomial moment matching method, along with recent results on low-rank approximation with single-vector Krylov methods. We use these results to show that the method can perform ‘implicit deflation’ as part of moment matching. 

Approximating the action of a matrix function $$f(\vec{A})$$ on a vector $$\vec{b}$$ is an increasingly important primitive in machine learning, data science, and statistics, with applications such as sampling high dimensional Gaussians, Gaussian process regression and Bayesian inference, principle component analysis, and approximating Hessian spectral densities. Over the past decade, a number of algorithms enjoying strong theoretical guarantees have been proposed for this task. Many of the most successful belong to a family of algorithms called \emph{Krylov subspace methods}. Remarkably, a classic Krylov subspace method, called the Lanczos method for matrix functions (Lanczos-FA), frequently outperforms newer methods in practice. Our main result is a theoretical justification for this finding: we show that, for a natural class of \emph{rational functions}, Lanczos-FA matches the error of the best possible Krylov subspace method up to a multiplicative approximation factor. The approximation factor depends on the degree of $f(x)$'s denominator and the condition number of $$\vec{A}$$, but not on the number of iterations $$k$$. Our result provides a strong justification for the excellent performance of Lanczos-FA, especially on functions that are well approximated by rationals, such as the matrix square root.