Sublinear Time Low-Rank Approximation of Hankel Matrices

2025-11-26

AI Summary

This work addresses the sublinear-time low-rank approximation of positive semidefinite (PSD) Hankel matrices. We propose the first structure-preserving and robust algorithm, leveraging Vandermonde matrix sampling and universal ridge leverage score bounds to achieve fast low-rank decomposition of large-scale PSD Hankel matrices. Theoretically, we establish—for the first time—the existence of a low-rank Hankel approximation satisfying the Beckermann–Townsend error bound, offering a novel finite-dimensional interpretation of the Adamyan–Arov–Krein (AAK) theorem. Algorithmically, our method outputs, in polylogarithmic time $mathrm{polylog}(n, 1/varepsilon)$, a rank-$O(log n cdot log(1/varepsilon))$ Hankel approximation $widehat{H}$ such that $|H - widehat{H}|_F leq O(|E|_F) + varepsilon |H|_F$, where $E$ denotes perturbation. By circumventing the standard $O(n^2)$ computational bottleneck, our approach significantly enhances scalability for Hankel matrices in signal processing and system identification.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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 $H in mathbb{R}^{n imes n}$ and any $ε> 0$, letting $H_k$ be the best rank-$k$ approximation of $H$, $|H-H_k|_F leq ε|H|_F$ for $k = O(log n log(1/ε))$. 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 $polylog(n, 1/ε)$ time, a factored representation of a rank-$O(log n log(1/ε))$ 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 $E in mathbb{R}^{n imes n}$ is an arbitrary non-Hankel noise matrix, we obtain error $|H - widehat{H}|_F leq O(|E|_F) + ε|H|_F$. 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.

Problem

• Developing sublinear time algorithm for Hankel matrix low-rank approximation
• Computing factored representation with polylogarithmic time complexity
• Achieving robust approximation under arbitrary non-Hankel noise

Innovation

• Sublinear time algorithm for Hankel low-rank approximation
• Structure-preserving existence of Hankel approximation
• Sampling-based approach using Vandermonde leverage scores