Sublinear Time Low-Rank Approximation of Hankel Matrices
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 ∈ Rn × n and any ε > 0, letting Hk be the best rank-k approximation of H, \|H-Hk\|F ≤ ε \|H\|F for k = O( n (1/ε)). As such, PSD Hankel matrices are natural targets for low-rank approximation algorithms. We give the first such algorithm that runs in sublinear time. In particular, we show how to compute, in (n, 1/ε) time, a factored representation of a rank-O( n (1/ε)) Hankel matrix H matching the error guarantee of Beckermann and Townsend up to constant factors. We further show that our algorithm is robust -- given input H+E where E ∈ Rn × n is an arbitrary non-Hankel noise matrix, we obtain error \|H - H\|F ≤ O(\|E\|F) + ε \|H\|F. Towards this algorithmic result, our first contribution is a structure-preserving existence result - we show that there exists a rank-k 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.
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