SPOQ ${\ell }_p$-Over-${\ell }_q$ Regularization for Sparse Signal Recovery applied to Mass Spectrometry

Abstract

Underdetermined or ill-posed inverse problems require additional information for \ldd{d} sound solutions with tractable optimization algorithms. Sparsity yields consequent heuristics to that matter, with numerous applications in signal restoration, image recovery, or machine learning. Since the ${\ell }_0$ count measure is barely tractable, many statistical or learning approaches have invested in computable proxies, such as the ${\ell }_1$ norm. However, the latter does not exhibit the desirable property of scale invariance for sparse data. Extending the SOOT Euclidean/Taxicab ${\ell }_1$-over-${\ell }_2$ norm-ratio initially introduced for blind deconvolution, we propose SPOQ, a family of smoothed (approximately) scale-invariant penalty functions. It consists of a Lipschitz-differentiable surrogate for ${\ell }_p$-over-${\ell }_q$ quasi-norm/norm ratios with $p\in ]0,2[$ and $q\ge 2$. This surrogate is embedded into a novel majorize-minimize trust-region approach, generalizing the variable metric forward-backward algorithm. For naturally sparse mass-spectrometry signals, we show that SPOQ significantly outperforms ${\ell }_0$, ${\ell }_1$, Cauchy, Welsch, SCAD and Celo penalties on several performance measures. Guidelines on SPOQ hyperparameters tuning are also provided, suggesting simple data-driven choices.

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Cited by in corpus (1)

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