Two talks that summarize my recent work
Online Robust PCA or Online Sparse + Low-Rank Matrix Recovery, Dept. Seminar in the ECE dept at CMU, February 2015
Also given in Communications and Signal Processing seminars at U. Michigan, Michigan State, Caltech, Rice, UT-Austin, TAMU, U. Iowa
Dynamic Structured Signals` Recovery and Applications in Bioimaging, Dept. Colloquium in the ECE dept at UIUC, December 2013
Also given at Northwestern University. Older versions of this talk also given at IISc Bangalore, UCSD, UCLA, UCSB, Caltech, U. Maryland, Northwestern Univ, Princeton.
Some key papers:
o Online Matrix Completion and Online Robust PCA, ISIT 2015 (this work strictly improves the results of the ICASSP 2015 paper listed below)
A Correctness Result for Online Robust PCA, ICASSP 2015
o Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise, IEEE Trans. Information Theory, August 2014
o An Online Algorithm for Separating Sparse and Low-dimensional Signal Sequences from their Sum, IEEE Trans. Signal Processing, August 2014, (older title: Practical ReProCS for Separating Sparse and Low-dimensional Signal Sequences from their Sum)
§ Introduced a novel online robust principal components' analysis (PCA) solution called Recursive Projected CS (ReProCS) that uses two extra but usually practically valid assumptions beyond those used by the batch methods - it needs accurate initial subspace knowledge and slow subspace change.
§ Online algorithms are needed for real-time applications; and even for offline applications, they are faster and need less storage compared to batch techniques. Moreover, online approaches can provide a natural way to exploit temporal dependencies in the dataset. We can show that ReProCS uses slow subspace change to provably allow for significantly more correlated support sets of the sparse vectors compared to some batch methods.
§ Application to the video analytics problems was demonstrated - we showed that ReProCS significantly outperforms most existing batch and online robust PCA algorithms for separating difficult video sequences into foreground and background layers. See webpages of Han Guo or Chenlu Qiu
§ Obtained a correctness result for ReProCS. We show that under mild assumptions, with high probability, ReProCS can exactly recover the support set of the sparse vectors at all times; and the reconstruction errors for both the sparse and low-dimensional vectors are bounded by a time-invariant and small value. Moreover, the subspace recovery error decays down to a small value within a short delay after a subspace change. To the best of our knowledge, our result is the first correctness result for an online (recursive) robust PCA algorithm or equivalently for sparse plus low-rank matrix recovery.