Optimal deep neural networks for sparse recovery via Laplace techniques

This paper introduces Laplace techniques for designing a neural network, with the goal of estimating simplex-constraint sparse vectors from compressed measurements. To this end, we recast the problem of MMSE estimation (w.r.t. a pre-defined uniform input distribution) as the problem of computing the centroid of some polytope that results from the intersection of the simplex and an affine subspace determined by the measurements. Owing to the specific structure, it is shown that the centroid can be computed analytically by extending a recent result that facilitates the volume computation of polytopes via Laplace transformations. A main insight of this paper is that the desired volume and centroid computations can be performed by a classical deep neural network comprising threshold functions, rectified linear (ReLU) and rectified polynomial (ReP) activation functions. The proposed construction of a deep neural network for sparse recovery is completely analytic so that time-consuming training procedures are not necessary. Furthermore, we show that the number of layers in our construction is equal to the number of measurements which might enable novel low-latency sparse recovery algorithms for a larger class of signals than that assumed in this paper. To assess the applicability of the proposed uniform input distribution, we showcase the recovery performance on samples that are soft-classification vectors generated by two standard datasets. As both volume and centroid computation are known to be computationally hard, the network width grows exponentially in the worst-case. It can be, however, decreased by inducing sparse connectivity in the neural network via a well-suited basis of the affine subspace. Finally, the presented analytical construction may serve as a viable initialization to be further optimized and trained using particular input datasets at hand.