Mathematics – Optimization and Control
Scientific paper
2009-06-11
SIAM Journal on Optimization, Vol. 21, Issue 2, pp. 572-596, 2011
Mathematics
Optimization and Control
Scientific paper
10.1137/090761793
Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is NP-hard in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the $\ell_1$ norm and the nuclear norm of the components. We develop a notion of \emph{rank-sparsity incoherence}, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature, with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems.
Chandrasekaran Venkat
Parrilo Pablo A.
Sanghavi Sujay
Willsky Alan S.
No associations
LandOfFree
Rank-Sparsity Incoherence for Matrix Decomposition does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Rank-Sparsity Incoherence for Matrix Decomposition, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Rank-Sparsity Incoherence for Matrix Decomposition will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-134582