In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. In this work, we study unfolded ISTA (Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery. We introduce a weight structure that is necessary for asymptotic convergence to the true sparse signal. With this structure, unfolded ISTA can attain a linear convergence, which is better than the sublinear convergence of ISTA FISTA in general cases. Furthermore, we propose to incorporate thresholding in the network to perform support selection, which is easy to implement and able to boost the convergence rate both theoretically and empirically. Extensive simulations, including sparse vector recovery and a compressive sensing experiment on real image data, corroborate our theoretical results and demonstrate their practical usefulness. We have made our codes publicly available: this https URL.
@cite_14: 
@cite_3: Deep learning has gained great popularity due to its widespread success on many inference problems. We consider the application of deep learning to the sparse linear inverse problem encountered in compressive sensing, where one seeks to recover a sparse signal from a small number of noisy linear measurements. In this paper, we propose a novel neural-network architecture that decouples prediction errors across layers in the same way that the approximate message passing (AMP) algorithm decouples them across iterations: through Onsager correction. Numerical experiments suggest that our “learned AMP” network significantly improves upon Gregor and Le-Cun's “learned ISTA” network in both accuracy and complexity.1
@cite_2: Compressive image recovery is a challenging problem that requires fast and accurate algorithms. Recently, neural networks have been applied to this problem with promising results. By exploiting massively parallel GPU processing architectures and oodles of training data, they can run orders of magnitude faster than existing techniques. However, these methods are largely unprincipled black boxes that are difficult to train and often-times specific to a single measurement matrix. It was recently demonstrated that iterative sparse-signal-recovery algorithms can be "unrolled" to form interpretable deep networks. Taking inspiration from this work, we develop a novel neural network architecture that mimics the behavior of the denoising-based approximate message passing (D-AMP) algorithm. We call this new network Learned D-AMP (LDAMP). The LDAMP network is easy to train, can be applied to a variety of different measurement matrices, and comes with a state-evolution heuristic that accurately predicts its performance. Most importantly, it outperforms the state-of-the-art BM3D-AMP and NLR-CS algorithms in terms of both accuracy and run time. At high resolutions, and when used with sensing matrices that have fast implementations, LDAMP runs over @math faster than BM3D-AMP and hundreds of times faster than NLR-CS.
In @cite_14 , a similar learning-based model inspired by another iterative algorithm solve LASSO, approximated message passing (AMP), was studied. The idea was advanced in @cite_2 to substituting the AMP proximal operator (soft-thresholding) with a learnable Gaussian denoiser. The resulting model, called Learned Denoising AMP (L-DAMP), has theoretical guarantees under the asymptotic assumption named state evolution.'' While the assumption is common in analyzing AMP algorithms, the tool is not directly applicable to ISTA. Moreover, @cite_14 shows L-DAMP is MMSE optimal, but there is no result on its convergence rate. Besides, we also note the empirical effort in @cite_3 that introduces an Onsager correction to LISTA to make it resemble AMP.