Low-Rank Approximation with $1/\epsilon^{1/3}$ Matrix-Vector Products

TL;DR

This paper introduces an improved algorithm for low-rank matrix approximation using Krylov methods that significantly reduces the number of matrix-vector products needed, achieving near-optimal complexity for Schatten-$p$ norms.

Contribution

The paper presents a novel Krylov-based algorithm that reduces matrix-vector product complexity for Schatten-$p$ norm low-rank approximation, extending results to all $p \, \geq 1$ and establishing tight lower bounds.

Findings

Uses only $\tilde{O}(k p^{1/6} / \epsilon^{1/3})$ matrix-vector products.

Improves previous bounds for $p=2$ from $\tilde{O}(k/\epsilon^{1/2})$ to $\tilde{O}(k/\epsilon^{1/3})$.

Establishes a lower bound of $\Omega(1/\epsilon^{1/3})$ for all fixed $p \geq 1$, showing optimality.

Abstract

We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Here, given access to a matrix $A$ through matrix-vector products, an accuracy parameter $\epsilon$, and a target rank $k$, the goal is to find a rank-$k$ matrix $Z$ with orthonormal columns such that $\| A(I -ZZ^\top)\|_{S_p} \leq (1+\epsilon)\min_{U^\top U = I_k} \|A(I - U U^\top)\|_{S_p}$, where $\|M\|_{S_p}$ denotes the $\ell_p$ norm of the the singular values of $M$. For the special cases of $p=2$ (Frobenius norm) and $p = \infty$ (Spectral norm), Musco and Musco (NeurIPS 2015) obtained an algorithm based on Krylov methods that uses $\tilde{O}(k/\sqrt{\epsilon})$ matrix-vector products, improving on the na\"ive $\tilde{O}(k/\epsilon)$ dependence obtainable by the power method, where $\tilde{O}$ suppresses poly$(\log(dk/\epsilon))$ factors. Our main result is an algorithm that uses only $\tilde{O}(kp^{1/6}/\epsilon^{1/3})$ matrix-vector products, and works for all $p \geq 1$. For $p = 2$ our bound improves the previous $\tilde{O}(k/\epsilon^{1/2})$ bound to $\tilde{O}(k/\epsilon^{1/3})$. Since the Schatten-$p$ and Schatten-$\infty$ norms are the same up to a $(1+ \epsilon)$-factor when $p \geq (\log d)/\epsilon$, our bound recovers the result of Musco and Musco for $p = \infty$. Further, we prove a matrix-vector query lower bound of $\Omega(1/\epsilon^{1/3})$ for any fixed constant $p \geq 1$, showing that surprisingly $\tilde{\Theta}(1/\epsilon^{1/3})$ is the optimal complexity for constant~$k$. To obtain our results, we introduce several new techniques, including optimizing over multiple Krylov subspaces simultaneously, and pinching inequalities for partitioned operators. Our lower bound for $p \in [1,2]$ uses the Araki-Lieb-Thirring trace inequality, whereas for $p>2$, we appeal to a norm-compression inequality for aligned partitioned operators.