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We address the problem of reconstructing sparse signals from noisy andcompressive measurements using a feed-forward deep neural network (DNN) with anarchitecture motivated by the iterative shrinkage-thresholding algorithm(ISTA). We maintain the weights and biases of the network links as prescribedby ISTA and model the nonlinear activation function using a linear expansion ofthresholds (LET), which has been very successful in image denoising anddeconvolution. The optimal set of coefficients of the parametrized activationis learned over a training dataset containing measurement-sparse signal pairs,corresponding to a fixed sensing matrix. For training, we develop an efficientsecond-order algorithm, which requires only matrix-vector product computationsin every training epoch (Hessian-free optimization) and offers superiorconvergence performance than gradient-descent optimization. Subsequently, wederive an improved network architecture inspired by FISTA, a faster version ofISTA, to achieve similar signal estimation performance with about 50% of thenumber of layers. The resulting architecture turns out to be a deep residualnetwork, which has recently been shown to exhibit superior performance inseveral visual recognition tasks. Numerical experiments demonstrate that theproposed DNN architectures lead to 3 to 4 dB improvement in the reconstructionsignal-to-noise ratio (SNR), compared with the state-of-the-art sparse codingalgorithms.

Deep Sparse Coding Using Optimized Linear Expansion of Thresholds