On the Null Space Constant for lp Minimization

1 Introduction The combinatorial optimization of $\ell_0$-minimization $$\label{l0min} \underset{\bf x}{\operatorname{argmin}}\|{\bf x}\|_0\ \ \textrm{subject to}\ \ {\bf Ax}={\bf y}$$ is NP-hard and therefore cannot be solved efficiently. A standard method to solve this problem is by relaxing the non-convex discontinuous $\ell_0$…

The Convergence Guarantees of a Non-convex Approach for Sparse Recovery

1 Introduction Since the introduction of compressive sensing, sparse recovery has received much attention and becomes a very hot topic these years. Sparse recovery aims to solve the underdetermined linear system ${\bf y}={\bf Ax}$, where ${\bf y}\in\mathbb{R}^M$ denotes the measurement…

On the Performance Bound of Sparse Estimation with Sensing Matrix Perturbation

1 Introduction The problem of sparse recovery from linear measurement has been a hot topic these years and has drawn a great deal of attention. Various practical algorithms of sparse recovery have been proposed and their performance in noisy scenarios…

Perturbation Analysis of Sparse Signal Recovery Algorithms

1 Introduction In the field of compressive sensing (CS), an $s$-sparse signal ${\bf x}\in\mathbb{R}^{n}$ is linearly observed by some certain known sensing matrix ${\bf A}\in\mathbb{R}^{m\times n}$ $${\bf y}={\bf Ax},$$ and the goal of sparse signal recovery is to…

Optical Analog-to-Digital Conversion System based on Compressive Sampling

1 Introduction With the development of radio frequency (RF) technology and digital signal processing, there has been an ever-growing demand for high-rate and high-precision analog-to-digital converters (ADCs). Due to the wide spectral that the multiband radio signals might lie in,…

Zero-point Attracting Projection (ZAP) —— an Excellent Non-convex Sparse Recovery Algorithm

1 Introduction Zero-point Attracting Projection (ZAP) is an excellent non-convex algorithm we proposed to solve the above sparse recovery problem. ZAP is first proposed in the paper A stochastic gradient approach on compressive sensing signal reconstruction based on adaptive filtering…

A Topological Viewpoint to Exact and Robust Sparse Recovery for $F$-minimization

1 Introduction This work Robustness of Sparse Recovery via F-minimization: A Topological Viewpoint by Jingbo Liu, Jian Jin, and Yuantao Gu has been published in IEEE Transactions on Information Theory, 61(7):3996-4014, 2015. Part of this work will be presented at…