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Sublinear Time Low-Rank Approximation of Hankel Matrices

Michael Kapralov, Cameron Musco, Kshiteej Sheth

TL;DR

This work proves that PSD Hankel matrices admit sublinear-time, structure-preserving low-rank approximations in the Frobenius norm with rank $O(\log n\log(1/\epsilon))$, matching the Beckermann–Townsend Frobenius guarantee. The key ideas combine a finite-dimensional analog of AAK theory with a log-scale bucket sparsification over Vandermonde nodes and universal ridge-leverage bounds for Vandermonde matrices, enabling a two-part decomposition $H=H_1+H_2$ and sublinear regression-based recovery. The authors also establish a near-tight lower bound on the approximate rank and show robustness to additive noise $E$, yielding $\|H-\widehat{H}\|_F \le C\|E\|_F + \epsilon\|H\|_F$. They develop a sublinear-time algorithm that recovers $H$ in a compressed Hankel form $VD'V^T+\widehat{H}'$, with $V$ Vandermonde and $D'$ diagonal, and demonstrate applications to polynomial basis transforms and Hankel covariance estimation. Overall, the results offer a path to fast, accurate, structure-preserving low-rank approximations for Hankel matrices with broad computational implications.

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 $ε> 0$, 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 $E \in \mathbb{R}^{n \times n}$ 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.

Sublinear Time Low-Rank Approximation of Hankel Matrices

TL;DR

This work proves that PSD Hankel matrices admit sublinear-time, structure-preserving low-rank approximations in the Frobenius norm with rank , matching the Beckermann–Townsend Frobenius guarantee. The key ideas combine a finite-dimensional analog of AAK theory with a log-scale bucket sparsification over Vandermonde nodes and universal ridge-leverage bounds for Vandermonde matrices, enabling a two-part decomposition and sublinear regression-based recovery. The authors also establish a near-tight lower bound on the approximate rank and show robustness to additive noise , yielding . They develop a sublinear-time algorithm that recovers in a compressed Hankel form , with Vandermonde and diagonal, and demonstrate applications to polynomial basis transforms and Hankel covariance estimation. Overall, the results offer a path to fast, accurate, structure-preserving low-rank approximations for Hankel matrices with broad computational implications.

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 and any , letting be the best rank- approximation of , for . 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 time, a factored representation of a rank- Hankel matrix matching the error guarantee of Beckermann and Townsend up to constant factors. We further show that our algorithm is \emph{robust} -- given input where is an arbitrary non-Hankel noise matrix, we obtain error . Towards this algorithmic result, our first contribution is a \emph{structure-preserving} existence result - we show that there exists a rank- \emph{Hankel} approximation to 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.

Paper Structure

This paper contains 25 sections, 26 theorems, 160 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Hankel Low-Rank Approximation.
  3. Our Contributions.
  4. Related Work.
  5. Paper Organization.
  6. Technical Overview.
  7. Existence of a Good Hankel Low Rank Approximation.
  8. Approximate Rank Lower Bound.
  9. Sublinear Time Robust Low Rank Approximation.
  10. Preliminaries and Notation.
  11. Notation
  12. Basic results regarding Hankel matrices.
  13. Linear algebra and polynomial sensitivity tools
  14. Existence of a Good Hankel Low Rank Approximation.
  15. Bucket Sparsification
  16. ...and 10 more sections

Key Result

Theorem 1

Let $H \in \mathbb{R}^{n\times n}$ be a PSD Hankel matrix, $E\in \mathbb{R}^{n\times n}$ be an arbitrary noise matrix, and $\epsilon > 0$ be an error parameter. Given entrywise access to $H+E$, alg:noisy_hankel_recovery runs in $\mathop{\mathrm{\mathrm{polylog}}}\limits( n,1/\epsilon)$ time and retu

Figures (2)

  • Figure 1: Sparsification of buckets $B_1,B_2$ to exponentiated Chebyshev nodes $T_1,T_2$ in respective intervals.
  • Figure 2: Entrywise upper bounds on $E_{0,1}+E_{1,1}+E_{2,1}+\ldots$, the shaded and unshaded region in each term are entrywise $1$ and $0$ respectively.

Theorems & Definitions (58)

  • Theorem 1: Sublinear Time Hankel Low-Rank Approximation
  • Theorem 2: Existence of Accurate Hankel Low-Rank Approximations
  • Theorem 3: Lower Bound on the Approximate Rank of PSD Hankel Matrices
  • Definition 2.1: Moment vector
  • Definition 2.2: Real Vandermonde matrix
  • Claim 2.3
  • Lemma 2.4: Fiedler Factorization
  • Lemma 2.5: Simplified, full version in \ref{['sec:bucket_sparsification']}
  • Lemma 2.6: Simplified, full version in \ref{['sec:prelim_hankel']}
  • Lemma 2.7: Simplified, full version in \ref{['sec:hankel_spectral_lower_bd']}
  • ...and 48 more