  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.
