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

Research Summary (new - updated
Dec8 2014)

**Talk
on Recursive Structured Signals Recovery
and Applications in Bio-Imaging, ECE
dept colloquium at UIUC, December 2013**

**Talk
on Online Robust PCA at UT Austin, Rice
and TAMU, November 2014**

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. Sig.
Proc., 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, Jinchun Zhan)- Modified-CS (IEEE Trans. Sig. Proc.,
Sept 2010)
- 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 (arXiv:1305.0842 [cs.IT], shorter versions in ISIT 2013,
Allerton 2010, revised and resubmitted to IEEE Trans. Info. Th.)
- Obtained mild 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. 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 A Correctness Result for Online Robust PCA, submitted, on arXiv:1409.3959 [cs.IT].

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 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.