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• W2119205172 abstract "Consider estimating a structured signal $mathbf{x}_0$ from linear, underdetermined and noisy measurements $mathbf{y}=mathbf{A}mathbf{x}_0+mathbf{z}$, via solving a variant of the lasso algorithm: $hat{mathbf{x}}=argmin_mathbf{x}{ |mathbf{y}-mathbf{A}mathbf{x}|_2+lambda f(mathbf{x})}$. Here, $f$ is a convex function aiming to promote the structure of $mathbf{x}_0$, say $ell_1$-norm to promote sparsity or nuclear norm to promote low-rankness. We assume that the entries of $mathbf{A}$ are independent and normally distributed and make no assumptions on the noise vector $mathbf{z}$, other than it being independent of $mathbf{A}$. Under this generic setup, we derive a general, non-asymptotic and rather tight upper bound on the $ell_2$-norm of the estimation error $|hat{mathbf{x}}-mathbf{x}_0|_2$. Our bound is geometric in nature and obeys a simple formula; the roles of $lambda$, $f$ and $mathbf{x}_0$ are all captured by a single summary parameter $delta(lambdapartial((f(mathbf{x}_0)))$, termed the Gaussian squared distance to the scaled subdifferential. We connect our result to the literature and verify its validity through simulations." @default.
• W2119205172 created "2016-06-24" @default.
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• W2119205172 date "2014-01-25" @default.
• W2119205172 modified "2023-09-27" @default.
• W2119205172 title "Simple Error Bounds for Regularized Noisy Linear Inverse Problems" @default.
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