Local recovery bounds for prior support constrained Compressed Sensing

Compressed Sensing with Nonlinear Observations and Related Nonlinear Optimisation Problems

Non-convex constraints have recently proven a valuable tool in many optimisation problems. In particular sparsity constraints have had a significant impact on sampling theory, where they are used in Compressed Sensing and allow structured signals to be sampled far below the rate traditionally prescribed. Nearly all of the theory developed for Compressed Sensing signal recovery assumes that samples are taken using linear measurements. In this paper we instead address the Compressed Sensing recovery problem in a setting where the observations are non-linear. We show that, under conditions simila…

Compressed Sensing from Phaseless Gaussian Measurements via Linear Programming in the Natural Parameter Space

We consider faithfully combining phase retrieval with classical compressed sensing. Inspired by the recent novel formulation for phase retrieval called PhaseMax, we present and analyze SparsePhaseMax, a linear program for phaseless compressed sensing in the natural parameter space. We establish that when provided with an initialization that correlates with an arbitrary $k$-sparse $n$-vector, SparsePhaseMax recovers this vector up to global sign with high probability from $O(k \log \frac{n}{k})$ magnitude measurements against i.i.d. Gaussian random vectors. Our proof of this fact exploits a cur…

Recovery of binary sparse signals from compressed linear measurements via polynomial optimization

The recovery of signals with finite-valued components from few linear measurements is a problem with widespread applications and interesting mathematical characteristics. In the compressed sensing framework, tailored methods have been recently proposed to deal with the case of finite-valued sparse signals. In this work, we focus on binary sparse signals and we propose a novel formulation, based on polynomial optimization. This approach is analyzed and compared to the state-of-the-art binary compressed sensing methods.

一种稀疏Kaczmarz求解器和线性化Bregman方法用于在线压缩感知

A sparse Kaczmarz solver and a linearized Bregman method for online compressed sensing

本文提出了一种计算稀疏或最小TV解的线性系统算法框架，包含Kaczmarz方法和线性化Bregman方法作为特例，并引入了稀疏Kaczmarz求解器等新方法，适用于测量缓慢且昂贵的在线压缩感知、TV断层成像和射电干涉测量等问题。

Non-convex approach to binary compressed sensing

We propose a new approach to the recovery of binary signals in compressed sensing, based on the local minimization of a non-convex cost functional. The desired signal is proved to be a local minimum of the functional under mild conditions on the sensing matrix and on the number of measurements. We develop a procedure to achieve the desired local minimum, and, finally, we propose numerical experiments that show the improvement obtained by the proposed approach with respect to the classical convex approach, i.e., Lasso.

A time continuation based fast approximate algorithm for compressed sensing related optimization

Recently, significant connections between compressed sensing problems and optimization of a particular class of functions relating to solutions of Hamilton-Jacobi equation was discovered. In this paper we introduce a fast approximate algorithm to optimize this particular class of functions and subsequently find the solution to the compressed sensing problem. Although we emphasize that the methodology of our algorithm finds an approximate solution, numerical experiments show that our algorithm perfectly recovers the solution when the solution is relatively sparse with respect to the number of m…