Improved RIP-Based Bounds for Guaranteed Performance of Two Compressed Sensing Algorithms

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two mainstream compressed sensing algorithms using the hard thresholding operator. The guaranteed performance of the two algorithms for signal recovery was mainly analyzed in terms of the restricted isometry property (RIP) of sensing matrices. At present, the best known bound using RIP of order … Read more

A new sufficient condition for non-convex sparse recovery via weighted $\ell_r\!-\!\ell_1$ minimization

In this letter, we discuss the reconstruction of sparse signals from undersampled data, which belongs to the core content of compressed sensing. A new sufficient condition in terms of the restricted isometry constant (RIC) and restricted orthogonality constant (ROC) is first established for the performance guarantee of recently proposed non-convex weighted $\ell_r-\ell_1$ minimization in recovering … Read more

Optimal K-Thresholding Algorithms for Sparse Optimization Problems

The simulations indicate that the existing hard thresholding technique independent of the residual function may cause a dramatic increase or numerical oscillation of the residual. This inherit drawback of the hard thresholding renders the traditional thresholding algorithms unstable and thus generally inefficient for solving practical sparse optimization problems. How to overcome this weakness and develop … Read more

Computing Restricted Isometry Constants via Mixed-Integer Semidefinite Programming

One of the fundamental tasks in compressed sensing is finding the sparsest solution to an underdetermined system of linear equations. It is well known that although this problem is NP-hard, under certain conditions it can be solved by using a linear program which minimizes the 1-norm. The restricted isometry property has been one of the … Read more

Sparse Recovery via Partial Regularization: Models, Theory and Algorithms

In the context of sparse recovery, it is known that most of existing regularizers such as $\ell_1$ suffer from some bias incurred by some leading entries (in magnitude) of the associated vector. To neutralize this bias, we propose a class of models with partial regularizers for recovering a sparse solution of a linear system. We … Read more

Sparse Recovery on Euclidean Jordan Algebras

We consider the sparse recovery problem on Euclidean Jordan algebra (SREJA), which includes sparse signal recovery and low-rank symmetric matrix recovery as special cases. We introduce the restricted isometry property, null space property (NSP), and $s$-goodness for linear transformations in $s$-sparse element recovery on Euclidean Jordan algebra (SREJA), all of which provide sufficient conditions for … Read more

Compressed Sensing: How sharp is the RIP?

Consider a measurement matrix A of size n×N, with n < N, y a signal in R^N, and b = Ay the observed measurement of the vector y. From knowledge of (b,A), compressed sensing seeks to recover the k-sparse x, k < n, which minimizes ||b-Ax||. Using various methods of analysis — convex polytopes, geometric … Read more

Phase Transitions for Greedy Sparse Approximation Algorithms

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 such algorithms have now been proven using the ubiquitous Restricted Isometry Property (RIP) [9] to have optimal-order uniform recovery guarantees. However, it is … Read more

Convergence of fixed-point continuation algorithms for matrix rank minimization

The matrix rank minimization problem has applications in many fields such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem. … Read more