We consider machine learning techniques to develop low-latency approximatesolutions to a class of inverse problems. More precisely, we use aprobabilistic approach for the problem of recovering sparse stochastic signalsthat are members of the $\ell_p$-balls. In this context, we analyze theBayesian mean-square-error (MSE) for two types of estimators: (i) a linearestimator and (ii) a structured estimator composed of a linear operatorfollowed by a Cartesian product of univariate nonlinear mappings. Byconstruction, the complexity of the proposed nonlinear estimator is comparableto that of its linear counterpart since the nonlinear mapping can beimplemented efficiently in hardware by means of look-up tables (LUTs). Theproposed structure lends itself to neural networks and iterativeshrinkage/thresholding-type algorithms restricted to a single iterate (e.g. dueto imposed hardware or latency constraints). By resorting to an alternatingminimization technique, we obtain a sequence of optimized linear operators andnonlinear mappings that converge in the MSE objective. The result is attractivefor real-time applications where general iterative and convex optimizationmethods are infeasible.
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