This paper studies the limiting behavior of Tyler’s and Maronna’s M-estimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples are identically and independently generated from the Gaussian distribution N(0, I), the difference between the sample covariance matrix and a scaled version of Tyler’s M-estimator or Maronna’s M-estimator tends to zero in spectral norm, and the empiri-cal spectral densities of both estimators converge to the Marchenko-Pastur distribution. We also extend this result to elliptical-distributed data sam-ples for Tyler’s M-estimator and non-isotropic Gaussian data samples for Maronna’s M-estimator. 1 ar

Abstract Non-negative matrix factorization (NMF) is a natural model of admixture and is widely used in science and engineering. A plethora of algorithms have been developed to tackle NMF, but due to the non-convex nature of the problem, there is little guarantee on how well these methods work. Recently a surge of research have focused on a very restricted class of NMFs, called separable NMF, where provably correct algorithms have been developed. In this paper, we propose the notion of subset-separable NMF, which substantially generalizes the property of separability. We show that subset-separability is a natural necessary condition for the factorization to be unique or to have minimum volume. We developed the Face-Intersect algorithm which provably and efficiently solves subset-separable NMF under natural conditions, and we prove that our algorithm is robust to small noise. We explored the performance of Face-Intersect on simulations and discuss settings where it empirically outperformed the state-of-art methods. Our work is a step towards finding provably correct algorithms that solve large classes of NMF problems.

This paper presents a fast algorithm for robust subspace recovery. The datasets considered include points drawn around a low-dimensional subspace of a higher dimensional ambient space, and a possibly large portion of points that do not lie nearby this subspace. The proposed algorithm, which we refer to as Fast Median Subspace (FMS), is designed to robustly determine the underlying subspace of such datasets, while having lower computational complexity than existing methods. Numerical experiments on synthetic and real data demonstrate its competitive speed and accuracy. 1