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
ℓ1 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
