Convolutional Sparse Support Estimator Network (CSEN): From Energy-Efficient Support Estimation to Learning-Aided Compressive Sensing

Abstract

Support estimation (SE) of a sparse signal refers to finding the location indices of the nonzero elements in a sparse representation. Most of the traditional approaches dealing with SE problems are iterative algorithms based on greedy methods or optimization techniques. Indeed, a vast majority of them use sparse signal recovery (SR) techniques to obtain support sets instead of directly mapping the nonzero locations from denser measurements (e.g., compressively sensed measurements). This study proposes a novel approach for learning such a mapping from a training set. To accomplish this objective, the convolutional sparse support estimator networks (CSENs), each with a compact configuration, are designed. The proposed CSEN can be a crucial tool for the following scenarios: 1) real-time and low-cost SE can be applied in any mobile and low-power edge device for anomaly localization, simultaneous face recognition, and so on and 2) CSEN's output can directly be used as "prior information," which improves the performance of sparse SR algorithms. The results over the benchmark datasets show that state-of-the-art performance levels can be achieved by the proposed approach with a significantly reduced computational complexity.

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“…Readers are referred to [9] for a more detailed literature review on SE and its applications. In the sequel, we briefly summarize the building blocks of the proposed approach.…”

“…Despite improved recognition accuracy, SRC solutions are iterative solutions and can be computationally demanding compared to CRC. In a recent work [9] , a compact NN design that can be considered as a bridge between NN-based and representation-based methodologies was proposed. The so-called CSEN uses a predefined dictionary and learns a direct mapping using moderate/low size training set, which maps query samples, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y}$ \end{document} , directly to the support set of representation coefficients, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {x}$ \end{document} (as it should be purely sparse in the ideal case).…”

“…However, many works in the literature dealing with SE problems tend to first apply a sparse recovery technique on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y}$ \end{document} to first get \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {\hat {x}}$ \end{document} , then use simple thresholding over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {\hat {x}}$ \end{document} to obtain a sparse SE, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\hat {\Lambda }$ \end{document} . However, SSR techniques such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\ell _{1}$ \end{document} -minimization are rather slow and their performance varies from one SRR tool to another [9] . In our previous work [9] , we proposed an alternative solution for this iterative sparse recovery approach which aims to learn a direct mapping from a test sample \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y}$ \end{document} to the corresponding support set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\hat {\Lambda }$ \end{document} .…”

“…However, SSR techniques such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\ell _{1}$ \end{document} -minimization are rather slow and their performance varies from one SRR tool to another [9] . In our previous work [9] , we proposed an alternative solution for this iterative sparse recovery approach which aims to learn a direct mapping from a test sample \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y}$ \end{document} to the corresponding support set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\hat {\Lambda }$ \end{document} . Along with the speed and stability compared to conventional SSR-based techniques and recent deep learning-based SSR solutions, CSEN has the crucial advantage of having a compact design that can achieve a good performance level even over scarce training data.…”

“…Therefore, given a predefined dictionary matrix, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {D}$ \end{document} and a test sample \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y}$ \end{document} , we expect the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {\hat {x}}$ \end{document} from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y} = \mathbf {D x}$ \end{document} , carry enough information about the class of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$\mathbf {y}$ \end{document} . Overall, in this study, we draw a convolutional support estimation network (CSEN) [9] -based solution pipeline, which fuses the representation-based classification scheme into a neural network (NN) body.…”