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 0 count measure is barely tractable, many statistical or learning approaches have invested in computable proxies, such as the 1 norm. However, the latter does not exhibit the desirable property of scale invariance for sparse data. Generalizing the SOOT Euclidean/Taxicab 1/ 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 p-over-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 forwardbackward algorithm. For naturally sparse mass-spectrometry signals, we show that SPOQ significantly outperforms 0, 1, Cauchy, Welsch, and CEL0 penalties on several performance measures. Guidelines on SPOQ hyperparameters tuning are also provided, suggesting simple data-driven choices.

GPU-based Implementations of MM Algorithms. Application to Spectroscopy Signal Restoration

“…where † denotes the pseudo-inverse operation. The computational cost necessary for one iteration is considerably reduced when compared to the HQ algorithm in (6). Note that the 3MG scheme is highly related to the nonlinear conjugate gradient algorithm [27], to the momentum-based accelerated gradient schemes from [28] and to trust-region approaches [29].…”

“…A first family of techniques amounts to minimizing a penalized loss function, combining data fidelity term and regularization term that incorporates prior knowledge (e.g., sparsity, positivity) on x. This approach was adopted, for instance in [3], [4] in the context of NMR relaxometry, in [5] in DOSY NMR, and in [6], [7] for MS data processing.…”

A Penalized Subspace Strategy for Solving Large-Scale Constrained Optimization Problems

“…To prevent this possible slowdown, we propose a modified strategy relying on local majorants rather than global ones. Such kind of local majorization strategy has already been successfully used for another MM algorithm in [26], leading to a significant speed up.…”

Unified Analysis on L1 over L2 Minimization for signal recovery

“…Email: xzhang@ee.ryerson.ca many applications. As such, various nonconvex minimizations have been developed due to their sharper approximation of L 0 norm [6][7][8].…”