Rank and null space calculations using matrix decomposition without column interchanges

The Column Space and the Null Space of a Matrix

Stable low-rank matrix recovery via null space properties

The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas such as quantum state tomography, machine learning and the PhaseLift approach to phaseless reconstruction problems. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically and convex optimization approaches inc...

Optimal column-based low-rank matrix reconstruction

We prove that for any real-valued matrix X ∈ R, and positive integers r > k, there is a subset of r columns of X such that projecting X onto their span gives a

New Null Space Results and Recovery Thresholds for Matrix Rank Minimization

Nuclear norm minimization (NNM) has recently gained significant attention for its use in rank minimization problems. Similar to compressed sensing, using null space characterizations, recovery thresholds for NNM have been studied in [12, 4]. However simulations show that the thresholds are far from optimal, especially in the low rank region. In this paper we apply the recent analysis of Stojnic...

Necessary and Sufficient Null Space Condition for Nuclear Norm Minimization in Low-Rank Matrix Recovery

Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is the nuclear norm minimization. In [12,14,15], the authors established the ne...