On robust recovery of signals from indirect observations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915141207982080 |
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| author | Bekri, Yannis Juditsky, Anatoli Nemirovski, Arkadi |
| author_facet | Bekri, Yannis Juditsky, Anatoli Nemirovski, Arkadi |
| contents | We consider an uncertain linear inverse problem as follows. Given observation $ω=Ax_*+ζ$ where $A\in {\bf R}^{m\times p}$ and $ζ\in {\bf R}^{m}$ 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 $ζ$ is "corrupted" -- contains an uncertain (deterministic dense or singular) component. Specifically, we assume that $ζ$ decomposes into $ζ=Nν_*+ξ$ where $ξ$ is the random noise and $Nν_*$ is the "adversarial contamination" with known $\cal N\subset {\bf R}^n$ such that $ν_*\in \cal N$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On robust recovery of signals from indirect observations Bekri, Yannis Juditsky, Anatoli Nemirovski, Arkadi Statistics Theory Machine Learning 62H12, 62J07, 62G08, 90C90 We consider an uncertain linear inverse problem as follows. Given observation $ω=Ax_*+ζ$ where $A\in {\bf R}^{m\times p}$ and $ζ\in {\bf R}^{m}$ 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 $ζ$ is "corrupted" -- contains an uncertain (deterministic dense or singular) component. Specifically, we assume that $ζ$ decomposes into $ζ=Nν_*+ξ$ where $ξ$ is the random noise and $Nν_*$ is the "adversarial contamination" with known $\cal N\subset {\bf R}^n$ such that $ν_*\in \cal N$ 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. |
| title | On robust recovery of signals from indirect observations |
| topic | Statistics Theory Machine Learning 62H12, 62J07, 62G08, 90C90 |
| url | https://arxiv.org/abs/2501.01935 |