FirmNet: A Sparsity Amplified Deep Network for Solving Linear Inverse Problems

Abstract

Recovering a sparse signal from a noisy linear measurement is an important problem in signal processing. Typically, one employs greedy pursuit techniques such as OMP, CoSaMP to solve an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> regularization problem. For large-scale problems, iterative shrinkage techniques such as ISTA, FISTA, AMP-ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> have been introduced. The underlying formulation in the iterative algorithms is a LASSO problem with an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -penalty. It is known in the literature that an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -penalty in LASSO suffers from underestimation of large signal amplitudes. Also, the iterative shrinkage-based approaches such as ISTA typically have only one free parameter to trade-off between noise variance and sparsity. We consider a minimax-concave penalty-based formulation, which offers an unbiased estimate of the sparse signal. The resulting iterative firm-thresholding algorithm is restructured as a DNN architecture called FirmNet. The proposed network, FirmNet, has two interpretable shrinkage function parameters - one that controls the noise variance, and the other that allows for explicit sparsity control. We compare the network with a broader network architecture of Learned-ISTA (LISTA), and show that it outperforms in terms of the probability-of-error-in-support (PES) - a strong support recovery metric, by at least three-fold. We also observe an improvement of 2 to 4 dB in reconstruction SNR compared with LISTA.