On the Nuclear Norm heuristic for a Hankel matrix Recovery Problem

TL;DR

This paper investigates the effectiveness of the nuclear norm heuristic in recovering impulse responses from Hankel matrices of stable single-pole systems, providing theoretical guarantees and exploring its application beyond specific cases.

Contribution

It offers a deterministic analysis and constructs a certificate for recovery, extending understanding of nuclear norm heuristic in structured matrix completion.

Findings

The nuclear norm heuristic can successfully recover certain impulse responses.

A certificate for guaranteed recovery is constructed based on matrix structure.

Experimental results support the theoretical insights and suggest broader applicability.

Abstract

This note addresses the question if and why the nuclear norm heuristic can recover an impulse response generated by a stable single-real-pole system, if elements of the upper-triangle of the associated Hankel matrix were given. Since the setting is deterministic, theories based on stochastic assumptions for low-rank matrix recovery do not apply here. A 'certificate' which guarantees the completion is constructed by exploring the structural information of the hidden matrix. Experimental results and discussions regarding the nuclear norm heuristic applied to a more general setting are also given.