On Robust Recovery of Signals from Indirect Observations

TL;DR

This paper develops and analyzes robust linear and polyhedral recovery algorithms for statistical inverse problems, effectively handling uncertainty in the observation matrix through convex optimization-based risk bounds.

Contribution

It introduces a framework for designing and analyzing robust estimates that account for matrix uncertainty using convex optimization, extending previous methods.

Findings

Risk bounds are effectively computed via convex optimization.

Robust estimates perform well under stochastic and deterministic uncertainties.

Optimization of risk bounds improves estimate robustness.

Abstract

Our focus is on robust recovery algorithms in statistical linear inverse problem. We consider two recovery routines - the much studied linear estimate originating from Kuks and Olman [42] and polyhedral estimate introduced in [37]. It was shown in [38] that risk of these estimates can be tightly upper-bounded for a wide range of a priori information about the model through solving a convex optimization problem, leading to a computationally efficient implementation of nearly optimal estimates of these types. The subject of the present paper is design and analysis of linear and polyhedral estimates which are robust with respect to the uncertainty in the observation matrix. We evaluate performance of robust estimates under stochastic and deterministic matrix uncertainty and show how the estimation risk can be bounded by the optimal value of efficiently solvable convex optimization problem; "presumably good" estimates of both types are then obtained through optimization of the risk bounds with respect to estimate parameters.