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- W3041234512 abstract "Compressed sensing involves solving a minimization problem with objective function $Omega(boldsymbol{x}) = |boldsymbol{x}|_1$ and linear constraints $boldsymbol{A} boldsymbol{x} = boldsymbol{b}$. Previous work has explored robustness to errors in $boldsymbol{A}$ and $boldsymbol{b}$ under special assumptions. Motivated by these results, we explore robustness to errors in $boldsymbol{A}$ for a wider class of objective functions $Omega$ and for a more general setting, where the solution may not be unique. Similar results for errors in $boldsymbol{b}$ are known and easier to prove. More precisely, for a seminorm $Omega(boldsymbol{x})$ with a polyhedral unit ball, we prove that the set-valued map $S(boldsymbol{A}) = arg min_{boldsymbol{A} boldsymbol{x} = boldsymbol{b}} Omega(boldsymbol{x})$ is calm in $boldsymbol{A}$, where calmness is a kind of local Lipschitz regularity." @default.
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- W3041234512 date "2020-07-11" @default.
- W3041234512 modified "2023-09-26" @default.
- W3041234512 title "You Need to Calm Down: Calmness Regularity for a Class of Seminorm Optimization Problems" @default.
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- W3041234512 doi "https://doi.org/10.48550/arxiv.2007.05689" @default.
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