On robust recovery of signals from indirect observations

TL;DR

This paper addresses the challenge of recovering signals from indirect observations contaminated with both random noise and adversarial corruption, proposing convex optimization methods for robust estimation in uncertain linear inverse problems.

Contribution

It introduces a framework for robust signal recovery under adversarial noise, extending uncertainty-immunized polyhedral estimates to ellitopic signal sets.

Findings

Effective convex optimization routines for robust estimation

Performance analysis of uncertainty-immunized estimates

Application to ellitopic signal sets

Abstract

We consider an uncertain linear inverse problem as follows. Given observation $\omega=Ax_*+\zeta$ where $A\in {\bf R}^{m\times p}$ and $\zeta\in {\bf R}^{m}$ is observation noise, we want to recover unknown signal $x_*$, known to belong to a convex set ${\cal X}\subset{\bf R}^{n}$. As opposed to the "standard" setting of such problem, we suppose that the model noise $\zeta$ is "corrupted" -- contains an uncertain (deterministic dense or singular) component. Specifically, we assume that $\zeta$ decomposes into $\zeta=N\nu_*+\xi$ where $\xi$ is the random noise and $N\nu_*$ is the "adversarial contamination" with known $\cal N\subset {\bf R}^n$ such that $\nu_*\in \cal N$ and $N\in {\bf R}^{m\times n}$. We consider two "uncertainty setups" in which $\cal N$ is either a convex bounded set or is the set of sparse vectors (with at most $s$ nonvanishing entries). We analyse the performance of "uncertainty-immunized" polyhedral estimates -- a particular class of nonlinear estimates as introduced in [15, 16] -- and show how "presumably good" estimates of the sort may be constructed in the situation where the signal set is an ellitope (essentially, a symmetric convex set delimited by quadratic surfaces) by means of efficient convex optimization routines.