This talk is about a curious phenomenon. Suppose we have a data
matrix, which is the superposition of a low-rank component and a
sparse component. Can we recover each component individually? We prove
that under some suitable assumptions, it is possible to recover both
the low-rank and the sparse components exactly by solving a very
convenient convex program called Principal Component Pursuit; among
all feasible decompositions, simply minimize a weighted combination of
the nuclear norm and of the L1 norm. This suggests the possibility of
a principled approach to robust principal component analysis since our
methodology and results assert that one can recover the principal
components of a data matrix even though a positive fraction of its
entries are arbitrarily corrupted. This extends to the situation where
a fraction of the entries are missing as well. We present applications
in the area of video surveillance, where our methodology allows for
the detection of objects in a cluttered background, and in the area of
face recognition, where it offers a principled way of removing shadows
and specularities in images of faces.