Identification of Sparse Linear Operators

TL;DR

This paper establishes conditions under which linear deterministic operators can be stably identified from their responses, showing that support area constraints enable reliable recovery without prior support knowledge.

Contribution

It introduces new theoretical bounds for operator identifiability based on support area, including the first provable algorithms for these cases.

Findings

Stable identifiability for support area D<=1/2.

Almost all operators are identifiable if D<1.

Algorithms are provided for operator recovery within these bounds.

Abstract

We consider the problem of identifying a linear deterministic operator from its response to a given probing signal. For a large class of linear operators, we show that stable identifiability is possible if the total support area of the operator's spreading function satisfies D<=1/2. This result holds for an arbitrary (possibly fragmented) support region of the spreading function, does not impose limitations on the total extent of the support region, and, most importantly, does not require the support region to be known prior to identification. Furthermore, we prove that stable identifiability of almost all operators is possible if D<1. This result is surprising as it says that there is no penalty for not knowing the support region of the spreading function prior to identification. Algorithms that provably recover all operators with D<=1/2, and almost all operators with D<1 are presented.