**Research summary and some important papers of Namrata Vaswani**

Research Summary (new - updated May,
2015)

**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:** **

**Recursive Recovery of Sparse Signal Sequences**- Kalman filtered Compressed Sensing (
*ICIP*2008) **First studied the problem of recursive reconstruction of sparse signal sequences with time-varying sparsity patterns**and proposed an efficient solution called KF-CS. Introduced the practically valid assumption of slow sparsity pattern change.- LS-CS-residual (
*IEEE Trans. Signal Processing*, Aug 2010) - The LS-CS-residual
idea and extensive simulations
- Bound for CS-residual
error and comparison with simple CS error bound
- Obtained mild
conditions under which the support errors and hence the signal
reconstruction errors of LS-CS are bounded by time-invariant and small
values (algorithm is stable)
- Brief idea of KF-CS
**Modified-CS: Sparse Recovery or Compressive Sensing with Partial Support Knowledge**(with Wei Lu or Jinchun Zhan)- Modified-CS
*(IEEE Trans. Signal Processing, Sept 2010), received the IEEE Signal Processing Society Best Paper Award for 2014* - The modified-CS idea,
extensive simulations and proof-of-concept applications in MRI/video
**Proof of Exact Reconstruction under much weaker assumptions**on the measurement matrix than those needed for simple CS (means: need much fewer measurements)- Regularized
modified-CS idea (use the prior knowledge of signal values along with
support knowledge)
- Time
Invariant Error Bounds for Modified-CS based Sparse Signal Sequence
Recovery,
*IEEE Trans. Information Theory*, March 2015 - Obtained conditions under which the support
errors and hence the signal reconstruction errors of modified-CS are
bounded by a time-invariant and small value at all times
(algorithm is ``stable``). Did this for a weaker and much more
practically valid signal change model than what was assumed in the LS-CS
paper. The exact same idea can be used to obtain similar conditions for
LS-CS as well.
**Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise**(with Chenlu Qiu, Brian Lois, Han Guo, Leslie Hogben)

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.*