Joint Sensing Matrix and Sparsifying Dictionary Optimization for Tensor Compressive Sensing.

Tensor compressive sensing (TCS) is a multidimensional framework of compressive sensing (CS), and it is advantageous in terms of reducing the amount of storage, easing hardware implementations, and preserving multidimensional structures of signals in comparison to a conventional CS system. In a TCS system, instead of using a random sensing matrix and a predefined dictionary, the average-case performance can be further improved by employing an optimized multidimensional sensing matrix and a learned multilinear sparsifying dictionary. In this paper, we propose an approach that jointly optimizes the sensing matrix and dictionary for a TCS system. For the sensing matrix design in TCS, an extended separable approach with a closed form solution and a novel iterative nonseparable method are proposed when the multilinear dictionary is fixed. In addition, a multidimensional dictionary learning method that takes advantages of the multidimensional structure is derived, and the influence of sensing matrices is taken into account in the learning process. A joint optimization is achieved via alternately iterating the optimization of the sensing matrix and dictionary. Numerical experiments using both synthetic data and real images demonstrate the superiority of the proposed approache

Tight-frame-like Sparse Recovery Using Non-tight Sensing Matrices

The choice of the sensing matrix is crucial in compressed sensing (CS). Gaussian sensing matrices possess the desirable restricted isometry property (RIP), which is crucial for providing performance guarantees on sparse recovery. Further, sensing matrices that constitute a Parseval tight frame result in minimum mean-squared-error (MSE) reconstruction given oracle knowledge of the support of the sparse vector. However, if the sensing matrix is not tight, could one achieve the reconstruction performance assured by a tight frame by suitably designing the reconstruction strategy? This is the key question that we address in this paper. We develop a novel formulation that relies on a generalized l2-norm-based data-fidelity loss that tightens the sensing matrix, along with the standard l1 penalty for enforcing sparsity. The optimization is performed using proximal gradient method, resulting in the tight-frame iterative shrinkage thresholding algorithm (TF-ISTA). We show that the objective convergence of TF-ISTA is linear akin to that of ISTA. Incorporating Nesterovs momentum into TF-ISTA results in a faster variant, namely, TF-FISTA, whose objective convergence is quadratic, akin to that of FISTA. We provide performance guarantees on the l2-error for the proposed formulation. Experimental results show that the proposed algorithms offer superior sparse recovery performance and faster convergence. Proceeding further, we develop the network variants of TF-ISTA and TF-FISTA, wherein a convolutional neural network is used as the sparsifying operator. On the application front, we consider compressed sensing image recovery (CSIR). Experimental results on Set11, BSD68, Urban100, and DIV2K datasets show that the proposed models outperform state-of-the-art sparse recovery methods, with performance measured in terms of peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM).Comment: 33 pages, 12 figure

Sparse Recovery Analysis of Preconditioned Frames via Convex Optimization

Orthogonal Matching Pursuit and Basis Pursuit are popular reconstruction algorithms for recovery of sparse signals. The exact recovery property of both the methods has a relation with the coherence of the underlying redundant dictionary, i.e. a frame. A frame with low coherence provides better guarantees for exact recovery. An equivalent formulation of the associated linear system is obtained via premultiplication by a non-singular matrix. In view of bounds that guarantee sparse recovery, it is very useful to generate the preconditioner in such way that the preconditioned frame has low coherence as compared to the original. In this paper, we discuss the impact of preconditioning on sparse recovery. Further, we formulate a convex optimization problem for designing the preconditioner that yields a frame with improved coherence. In addition to reducing coherence, we focus on designing well conditioned frames and numerically study the relationship between the condition number of the preconditioner and the coherence of the new frame. Alongside theoretical justifications, we demonstrate through simulations the efficacy of the preconditioner in reducing coherence as well as recovering sparse signals.Comment: 9 pages, 5 Figure