It is known that a high-dimensional sparse vector x∗ in R can be recovered from low-dimensional measurements y = Ax∗ where Am×n(m < n) is the measurement matrix. In this paper, we investigate the recovering ability of `p-minimization (0 ≤ p ≤ 1) as p varies, where `p-minimization returns a vector with the least `p “norm” among all the vectors x satisfying Ax = y. Besides analyzing the performan...

It is known that a high-dimensional sparse vector x∗ in Rn can be recovered from low-dimensional measurements y = Ax∗ where Am×n(m < n) is the measurement matrix. In this paper, with A being a random Gaussian matrix, we investigate the recovering ability of `p-minimization (0 ≤ p ≤ 1) as p varies, where `p-minimization returns a vector with the least `p quasi-norm among all the vectors x satisf...

Sparse approximate inverse (SAI) techniques have recently emerged as a new class of parallel preconditioning techniques for solving large sparse linear systems on high performance computers. The choice of the sparsity pattern of the SAI matrix is probably the most important step in constructing an SAI preconditioner. Both dynamic and static sparsity pattern selection approaches have been propos...

The dependence structure of extreme events of multivariate nature plays a special role for risk management applications, in particular in hydrology (flood risk). In a high dimensional context (d > 50), a natural first step is dimension reduction. Analyzing the tails of a dataset requires specific approaches: earlier works have proposed a definition of sparsity adapted for extremes, together wit...

Distributed Compressive Sensing (DCS) studies the recovery of jointly sparse signals. Compared to separate recovery, the joint recovery algorithms in DCS are usually more effective as they make use of the joint sparsity. In this thesis, we study a weighted l1-minimization algorithm for the joint sparsity model JSM-1 proposed by Baron et al. Our analysis gives a sufficient null space property fo...

This paper presents a su cient condition on sparsity patterns for the existence of the incomplete Cholesky factorization. Given the sparsity pattern P (A) of a matrix A, and a target sparsity pattern P satisfying the condition, incomplete Cholesky factorization successfully completes for all symmetric positive de nite matrices with the same pattern P (A). It is also shown that this condition is...

This paper presents a suucient condition on sparsity patterns for the existence of the incomplete Cholesky factorization. Given the sparsity pattern P(A) of a matrix A, and a target sparsity pattern P satisfying the condition, incomplete Cholesky factorization successfully completes for all symmetric positive deenite matrices with the same pattern P(A). This condition is also necessary in the s...

Compressed sensing deals with the reconstruction of sparse signals using a small number of linear measurements. One of the main challenges in compressed sensing is to find the support of a sparse signal. In the literature, several bounds on the scaling law of the number of measurements for successful support recovery have been derived where the main focus is on random Gaussian measurement matri...

Linear sketching and recovery of sparse vectors with randomly constructed sparse matrices has numerous applications in several areas, including compressive sensing, data stream computing, graph sketching, and combinatorial group testing. This paper considers the same problem with the added twist that the sparse coefficients of the unknown vector exhibit further correlations as determined by a k...

In this paper, we develop verifiable and computable performance analysis of sparsity recovery. We define a family of goodness measures for arbitrary sensing matrices as a set of optimization problems, and design algorithms with a theoretical global convergence guarantee to compute these goodness measures. The proposed algorithms solve a series of second-order cone programs, or linear programs. ...