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 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. Generalizing the SOOT Euclidean/Taxicab \(\ell_1/\ell_2\) norm-ratio initially introduced for blind deconvolution, we propose SPOQ, a family of smoothed 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, and \celo penalties on several performance measures. Guidelines on SPOQ hyperparameters tuning are also provided, suggesting simple data-driven choices.