Sublinear Time Low-Rank Approximation of Hankel Matrices

TL;DR

This paper introduces the first sublinear time algorithm for low-rank approximation of PSD Hankel matrices, leveraging their structure to achieve efficient, robust approximations with theoretical guarantees.

Contribution

It presents a novel sublinear time algorithm for low-rank Hankel approximation, along with a structure-preserving existence proof and a sampling-based approach utilizing Vandermonde matrix properties.

Findings

Achieves sublinear time complexity for low-rank Hankel approximation.

Provides robustness to noise in the input matrix.

Matches the error bounds of previous methods up to constant factors.

Abstract

Hankel matrices are an important class of highly-structured matrices, arising across computational mathematics, engineering, and theoretical computer science. It is well-known that positive semidefinite (PSD) Hankel matrices are always approximately low-rank. In particular, a celebrated result of Beckermann and Townsend shows that, for any PSD Hankel matrix $H \in \mathbb{R}^{n \times n}$ and any $\epsilon > 0$, letting $H_k$ be the best rank-$k$ approximation of $H$, $\|H-H_k\|_F \leq \epsilon \|H\|_F$ for $k = O(\log n \log(1/\epsilon))$. As such, PSD Hankel matrices are natural targets for low-rank approximation algorithms. We give the first such algorithm that runs in \emph{sublinear time}. In particular, we show how to compute, in $\polylog(n, 1/\epsilon)$ time, a factored representation of a rank-$O(\log n \log(1/\epsilon))$ Hankel matrix $\widehat{H}$ matching the error guarantee of Beckermann and Townsend up to constant factors. We further show that our algorithm is \emph{robust} -- given input $H+E$ where $E \in \mathbb{R}^{n \times n}$ is an arbitrary non-Hankel noise matrix, we obtain error $\|H - \widehat{H}\|_F \leq O(\|E\|_F) + \epsilon \|H\|_F$. Towards this algorithmic result, our first contribution is a \emph{structure-preserving} existence result - we show that there exists a rank-$k$ \emph{Hankel} approximation to $H$ matching the error bound of Beckermann and Townsend. Our result can be interpreted as a finite-dimensional analog of the widely applicable AAK theorem, which shows that the optimal low-rank approximation of an infinite Hankel operator is itself Hankel. Armed with our existence result, and leveraging the well-known Vandermonde structure of Hankel matrices, we achieve our sublinear time algorithm using a sampling-based approach that relies on universal ridge leverage score bounds for Vandermonde matrices.