Convex Reconstruction of Structured Matrix Signals from Linear Measurements:Theoretical Results

• 作者: Yuan Tian
• 来源期刊: 国际计算机前沿大会会议论文集
• 学科: 数学
• 关键词: Compressive sensing Structured matrix signal Convex optimization Column-wise sparsity FLATNESS Sign-stability Support-stability Robustness Random measurement
• 年，卷（期）: 2020,（1）
• 页码范围: 189-221
• 页数: 33页
• 分类号: O17

摘要

The problem of reconstructing n-by-n structured matrix signal X=(x1,...,xn)via convex optimization is investigated,where each column xj is a vector of s-sparsity and all columns have the same l1-norm value.In this paper,the convex programming problem was solved with noise-free or noisy measurements.The uniform sufficient conditions were established which are very close to necessary conditions and non-uniform conditions were also discussed.In addition,stronger conditions were investigated to guarantee the reconstructed signal’s support stability,sign stability and approximation-error robustness.Moreover,with the convex geometric approach in random measurement setting,one of the critical ingredients in this contribution is to estimate the related widths’bounds in case of Gaussian and non-Gaussian distributions.These bounds were explicitly controlled by signal’s structural parameters r and s which determined matrix signal’s column-wise sparsity and l1-column-flatness respectively.This paper provides a relatively complete theory on column-wise sparse and l1-column-flat matrix signal reconstruction,as well as a heuristic foundation for dealing with more complicated high-order tensor signals in,e.g.,statistical big data analysis and related data-intensive applications.