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Underdetermined or ill-posed inverse problems require additional information for 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 ℓ_0 count measure is barely tractable, many statistical or learning approaches have invested in computable proxies, such as the ℓ_1 norm. However, the latter does not exhibit the desirable property of scale<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2001.08496v2">arXiv:2001.08496v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tqkyhud34rgpvhvcixschcw6au">fatcat:tqkyhud34rgpvhvcixschcw6au</a> </span>
more »... ariance for sparse data. Extending the SOOT Euclidean/Taxicab ℓ_1-over-ℓ_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 ℓ_p-over-ℓ_q 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 ℓ_0, ℓ_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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