10,386 research outputs found
Optimal Column-Based Low-Rank Matrix Reconstruction
We prove that for any real-valued matrix , and
positive integers , there is a subset of columns of such that
projecting onto their span gives a -approximation
to best rank- approximation of in Frobenius norm. We show that the
trade-off we achieve between the number of columns and the approximation ratio
is optimal up to lower order terms. Furthermore, there is a deterministic
algorithm to find such a subset of columns that runs in arithmetic operations where is the exponent of matrix
multiplication. We also give a faster randomized algorithm that runs in arithmetic operations.Comment: 8 page
Unicity conditions for low-rank matrix recovery
Low-rank matrix recovery addresses the problem of recovering an unknown
low-rank matrix from few linear measurements. Nuclear-norm minimization is a
tractible approach with a recent surge of strong theoretical backing. Analagous
to the theory of compressed sensing, these results have required random
measurements. For example, m >= Cnr Gaussian measurements are sufficient to
recover any rank-r n x n matrix with high probability. In this paper we address
the theoretical question of how many measurements are needed via any method
whatsoever --- tractible or not. We show that for a family of random
measurement ensembles, m >= 4nr - 4r^2 measurements are sufficient to guarantee
that no rank-2r matrix lies in the null space of the measurement operator with
probability one. This is a necessary and sufficient condition to ensure uniform
recovery of all rank-r matrices by rank minimization. Furthermore, this value
of precisely matches the dimension of the manifold of all rank-2r matrices.
We also prove that for a fixed rank-r matrix, m >= 2nr - r^2 + 1 random
measurements are enough to guarantee recovery using rank minimization. These
results give a benchmark to which we may compare the efficacy of nuclear-norm
minimization
- …