Related papers: Average-case Hardness of RIP Certification
It is now well known that sparse or compressible vectors can be stably recovered from their low-dimensional projection, provided the projection matrix satisfies a Restricted Isometry Property (RIP). We establish new implications of the RIP…
The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound…
Orthogonal Matching Pursuit (OMP) has long been considered a powerful heuristic for attacking compressive sensing problems; however, its theoretical development is, unfortunately, somewhat lacking. This paper presents an improved Restricted…
We formulate a generalization of the Restricted Isometry Property (RIP) referred to as the Restricted Quasiconvexity Isometry Property (RQIP) for alpha stable random projections with $0<\alpha<1$. A lower bound on the number of rows for…
A major enterprise in compressed sensing and sparse approximation is the design and analysis of computationally tractable algorithms for recovering sparse, exact or approximate, solutions of underdetermined linear systems of equations. Many…
Consider the problem of recovering an unknown signal from undersampled measurements, given the knowledge that the signal has a sparse representation in a specified dictionary $D$. This problem is now understood to be well-posed and…
We study constructions of $k \times n$ matrices $A$ that both (1) satisfy the restricted isometry property (RIP) at sparsity $s$ with optimal parameters, and (2) are efficient in the sense that only $O(n\log n)$ operations are required to…
This paper provides a new tractable lower bound for the sparse recovery threshold of sensing matrices. This lower bound is used as a proxy to quantify the quality of sensing matrices in two different applications. First, it serves as…
Compressed sensing is a technique for finding sparse solutions to underdetermined linear systems. This technique relies on properties of the sensing matrix such as the restricted isometry property. Sensing matrices that satisfy this…
Inspired by significant real-life applications, in particular, sparse phase retrieval and sparse pulsation frequency detection in Asteroseismology, we investigate a general framework for compressed sensing, where the measurements are…
We consider the problem of recovering sparse vectors from underdetermined linear measurements via $\ell_p$-constrained basis pursuit. Previous analyses of this problem based on generalized restricted isometry properties have suggested that…
This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted…
Online sparse linear regression is an online problem where an algorithm repeatedly chooses a subset of coordinates to observe in an adversarially chosen feature vector, makes a real-valued prediction, receives the true label, and incurs the…
Dimensionality reduction is a popular approach to tackle high-dimensional data with low-dimensional nature. Subspace Restricted Isometry Property, a newly-proposed concept, has proved to be a useful tool in analyzing the effect of…
Random matrices are widely used in sparse recovery problems, and the relevant properties of matrices with i.i.d. entries are well understood. The current paper discusses the recently introduced Restricted Eigenvalue (RE) condition, which is…
We make a trivial modification to the elegant analysis of Garg and Khandekar (\emph{Gradient Descent with Sparsification} ICML 2009) that replaces the standard Restricted Isometry Property (RIP), with another RIP-type property (which could…
Signal models formed as linear combinations of few atoms from an over-complete dictionary or few frame vectors from a redundant frame have become central to many applications in high dimensional signal processing and data analysis. A core…
In the context of compressed sensing (CS), both Subspace Pursuit (SP) and Compressive Sampling Matching Pursuit (CoSaMP) are very important iterative greedy recovery algorithms which could reduce the recovery complexity greatly comparing…
The fields of compressed sensing (CS) and matrix completion have shown that high-dimensional signals with sparse or low-rank structure can be effectively projected into a low-dimensional space (for efficient acquisition or processing) when…
We study a class of real robust phase retrieval problems under a Gaussian assumption on the coding matrix when the received signal is sparsely corrupted by noise. The goal is to establish conditions on the sparsity under which the input…