Compressive Identification of Linear Operators

TL;DR

This paper establishes conditions under which linear operators can be stably identified from input-output data, revealing that support area constraints enable identification without prior support knowledge.

Contribution

It introduces new bounds on the support area of the spreading function for stable identifiability of linear operators, even with fragmented support regions and unknown support.

Findings

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

Almost all operators are identifiable if support area D <= 1.

No prior knowledge of support region needed for identifiability.

Abstract

We consider the problem of identifying a linear deterministic operator from an input-output measurement. For the large class of continuous (and hence bounded) operators, under additional mild restrictions, 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 arbitrary (possibly fragmented) support regions 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 of the spreading function to be known prior to identification. Furthermore, we prove that asking for identifiability of only almost all operators, stable identifiability 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.