Towards optimal nonlinearities for sparse recovery using higher-order statistics

TL;DR

This paper develops a probabilistic machine learning approach for sparse signal recovery, introducing a structured nonlinear estimator optimized via alternating minimization, suitable for low-latency, real-time applications.

Contribution

It proposes a novel structured nonlinear estimator with optimized linear and nonlinear components for sparse recovery, efficient for hardware implementation and real-time use.

Findings

The nonlinear estimator achieves lower MSE than linear estimators.

Optimized components converge via alternating minimization.

Suitable for hardware-constrained, low-latency environments.

Abstract

We consider machine learning techniques to develop low-latency approximate solutions to a class of inverse problems. More precisely, we use a probabilistic approach for the problem of recovering sparse stochastic signals that are members of the $\ell_p$-balls. In this context, we analyze the Bayesian mean-square-error (MSE) for two types of estimators: (i) a linear estimator and (ii) a structured estimator composed of a linear operator followed by a Cartesian product of univariate nonlinear mappings. By construction, the complexity of the proposed nonlinear estimator is comparable to that of its linear counterpart since the nonlinear mapping can be implemented efficiently in hardware by means of look-up tables (LUTs). The proposed structure lends itself to neural networks and iterative shrinkage/thresholding-type algorithms restricted to a single iterate (e.g. due to imposed hardware or latency constraints). By resorting to an alternating minimization technique, we obtain a sequence of optimized linear operators and nonlinear mappings that converge in the MSE objective. The result is attractive for real-time applications where general iterative and convex optimization methods are infeasible.