This work studies two interrelated problems - online robust PCA (RPCA) and
online low-rank matrix completion (MC). In recent work by Cand\`{e}s et al.,
RPCA has been defined as a problem of separating a low-rank matrix (true data),
L:=[β1β,β2β,β¦βtβ,β¦,βtmaxββ] and a sparse
matrix (outliers), S:=[x1β,x2β,β¦xtβ,β¦,xtmaxββ] from their
sum, M:=L+S. Our work uses this definition of RPCA. An important application
where both these problems occur is in video analytics in trying to separate
sparse foregrounds (e.g., moving objects) and slowly changing backgrounds.
While there has been a large amount of recent work on both developing and
analyzing batch RPCA and batch MC algorithms, the online problem is largely
open. In this work, we develop a practical modification of our recently
proposed algorithm to solve both the online RPCA and online MC problems. The
main contribution of this work is that we obtain correctness results for the
proposed algorithms under mild assumptions. The assumptions that we need are:
(a) a good estimate of the initial subspace is available (easy to obtain using
a short sequence of background-only frames in video surveillance); (b) the
βtβ's obey a `slow subspace change' assumption; (c) the basis vectors for
the subspace from which βtβ is generated are dense (non-sparse); (d) the
support of xtβ changes by at least a certain amount at least every so often;
and (e) algorithm parameters are appropriately setComment: Presented at ISIT (IEEE Intnl. Symp. on Information Theory), 2015.
Submitted to IEEE Transactions on Information Theory. This version: changes
are in blue; the main changes are just to explain the model assumptions
better (added based on ISIT reviewers' comments