Compressive principal component pursuit

Abstract

We consider the problem of recovering a target matrix that is a superposition of low-rank and sparse components, from a small set of linear measurements. This problem arises in compressed sensing of structured high-dimensional signals such as videos and hyperspectral images, as well as in the analysis of transformation invariant low-rank recovery. We analyze the performance of the natural convex heuristic for solving this problem, under the assumption that measurements are chosen uniformly at random. We prove that this heuristic exactly recovers low-rank and sparse terms, provided the number of observations exceeds the number of intrinsic degrees of freedom of the component signals by a polylogarithmic factor. Our analysis introduces several ideas that may be of independent interest for the more general problem of compressive sensing of superpositions of structured signals. 1

Compressive Online Video Background-Foreground Separation Using Multiple Prior Information and Optical Flow

“…Incremental PCP [9] processes each column-vector in M at a time, but it needs access to complete data (e.g., full frames) rather than compressive data. A counterpart of batch RPCA that operates on compressive measurements known as Compressive PCP can be found in [16]. The studies in [12][13][14][15]17] aim at solving the problem of online estimation of low-dimensional subspaces from randomly subsampled data for modeling the background.…”

Stable Analysis of Compressive Principal Component Pursuit

“…It is more important that they proved that these two components can be recovered by solving a simple convex optimization problem. In [11], John Wright et al generalized this problem to decompose a matrix into multiple incoherent components:…”

“…The authors also provide a sufficient condition that can promise the existence and uniqueness theorem of compressive principle component pursuit (CPCP). The result in [11] requires that the components are low-complexity structures.…”

“…In order to further complete the theory developed in [11], it is necessary to research the stability of CPCP which can guarantee stable and accurate recovery in the presence of entry-wise noise. In this paper, we make a commendable attempt in this respect.…”

Separating sparse and low-dimensional signal sequences from time-varying undersampled projections of their sums

“…In very recent works [16,17,18] the compressive robust PCA problem has also been looked at. All these works assume that the low dimensional signal sequence is dense (resulting in a low-rank but dense matrix) and the sparse signal sequence has independent and identically distributed (i.i.d.)…”