Matrix Sparsification and the Sparse Null Space Problem

The Column Space and the Null Space of a Matrix

The null-space method and its relationship with matrix factorizations for sparse saddle point systems

The null-space method for solving saddle point systems of equations has long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified the equivalence of the null-space method and matrix factorizations. In this report, we review these findings, highlight links between them, and bring t...

Iterative Null-space Projection Method with Adaptive Thresholding in Sparse Signal Recovery and Matrix Completion

Adaptive thresholding methods have proved to yield high Signal-to-Noise Ratio (SNR) and fast convergence in finding the solution to the Compressed Sensing (CS) problems. Recently, It was observed that the robustness of a class of iterative sparse recovery algorithms such as Iterative Method with Adaptive Thresholding (IMAT) outperforms the well-known LASSO algorithm in terms of reconstruction q...

Solving a Tropical Optimization Problem via Matrix Sparsification

Optimization problems are considered in the framework of tropical algebra to minimize and maximize a nonlinear objective function defined on vectors over an idempotent semifield, and calculated using multiplicative conjugate transposition. To find the minimum of the function, we first obtain a partial solution, which explicitly represents a subset of solution vectors. We characterize all soluti...

Random Search Algorithms for the Sparse Null Vector Problem

We consider the following problem: Given a matrix A, find minimal subsets of columns of A with cardinality no larger than a given bound that are linear dependent or nearly so. This problem arises in various forms in optimization, electrical engineering, and statistics. In its full generality, the problem is known to be NP-complete. We present a Monte Carlo method that finds such subsets with hi...