On robust recovery of signals from indirect observations

• URL: http://arxiv.org/abs/2501.01935v2
• Date: Thu, 06 Feb 2025 17:15:22 GMT
• Title: On robust recovery of signals from indirect observations
• Authors: Yannis Bekri, Anatoli Juditsky, Arkadi Nemirovski,
• Abstract summary: We show how "presumably good" estimates of the sort may be constructed in the situation where the signal set is a convex set. We consider two "uncertainty setups" in which $cal N$ is either a convex bounded set or is the set of sparse vectors.
• Score: 0.24578723416255752
• License:
• 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.

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