SPOQ: smooth, sparse ℓp-over-ℓq ratio regularization toolbox

Sparsity-promoting data restoration/recovery with SPOQ smooth/non-convex penalty with quasi-norm/norm ratios to emulate the ℓ0 count measure

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 ℓ0 count measure is barely tractable, many statistical or learning approaches have invested in computable proxies, such as the ℓ1 norm. However, the latter does not exhibit the desirable property of scale invariance for sparse data. Extending the SOOT Euclidean/Taxicab ℓ1-over-ℓ2 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 ℓp-over-ℓq quasi-norm/norm ratios with p∈]0,2[ and q≥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 ℓ0, ℓ1, Cauchy, Welsch, SCAD and Cel0 penalties on several performance measures. Guidelines on SPOQ hyperparameters tuning are also provided, suggesting simple data-driven choices.