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Wednesday, February 20, 2013

Compressive Principal Component Pursuit

Here are two presentations of the Compressive Principal Component Pursuit and Modeling High-dimensional (Visual) Data by Yi Ma which feature in part some grounding theoretical work to PCP (  for instance SpaRCS: is based on CoSAMP which competes with CPCP, yet has little theoretical justification)

The supporting papers which provide the theoretical grounding are: Principal Component Pursuit with Reduced Linear Measurements by Arvind Ganesh, Kerui Min, John Wright, Yi Ma

In this paper, we study the problem of decomposing a superposition of a low-rank matrix and a sparse matrix when a relatively few linear measurements are available. This problem arises in many data processing tasks such as aligning multiple images or rectifying regular texture, where the goal is to recover a low-rank matrix with a large fraction of corrupted entries in the presence of nonlinear domain transformation. We consider a natural convex heuristic to this problem which is a variant to the recently proposed Principal Component Pursuit. We prove that under suitable conditions, this convex program guarantees to recover the correct low-rank and sparse components despite reduced measurements. Our analysis covers both random and deterministic measurement models.
and Compressive Principal Component Pursuit by John Wright, Arvind Ganesh, Kerui Min, Yi Ma
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 compressed sensing and decomposing superpositions of multiple structured signals.

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