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Recursive robust pca or recursive sparse recovery in large but structured noise”, arXiv: 1211.3754 [cs.IT
, 2012
"... We study the recursive robust principal components ’ analysis (PCA) problem. Here, “robust ” refers to robustness to both independent and correlated sparse outliers. If the outlier is the signalofinterest, this problem can be interpreted as one of recursively recovering a time sequence of sparse v ..."
Abstract

Cited by 3 (3 self)
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We study the recursive robust principal components ’ analysis (PCA) problem. Here, “robust ” refers to robustness to both independent and correlated sparse outliers. If the outlier is the signalofinterest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt: the noise needs to lie in a “slowly changing ” low dimensional subspace. We study a novel solution called Recursive Projected CS (ReProCS). Under mild assumptions, we show that, with high probability (w.h.p.), at all times, ReProCS can exactly recover the support set of St; and the reconstruction errors of both St andLt are upper bounded by a timeinvariant and small value. Index Terms — robust PCA, compressive sensing 1.
Recursive sparse recovery in large but structured noise  part 2,” arXiv: 1211.3754 [cs.IT
, 2013
"... Abstract—We study the problem of recursively recovering a time sequence of sparse vectors, St, from measurements Mt: = St + Lt that are corrupted by structured noise Lt which is dense and can have large magnitude. The structure that we require is that Lt should lie in a low dimensional subspace that ..."
Abstract

Cited by 2 (2 self)
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Abstract—We study the problem of recursively recovering a time sequence of sparse vectors, St, from measurements Mt: = St + Lt that are corrupted by structured noise Lt which is dense and can have large magnitude. The structure that we require is that Lt should lie in a low dimensional subspace that is either fixed or changes “slowly enough”; and the eigenvalues of its covariance matrix are “clustered”. We do not assume any model on the sequence of sparse vectors. Their support sets and their nonzero element values may be either independent or correlated over time (usually in many applications they are correlated). The only thing required is that there be some support change every so often. We introduce a novel solution approach called Recursive Projected Compressive Sensing with clusterPCA (ReProCScPCA) that addresses some of the limitations of earlier work. Under mild assumptions, we show that, with high probability, ReProCScPCA can exactly recover the support set of St at all times; and the reconstruction errors of both St and Lt are upper bounded by a timeinvariant and small value. I.
SEPARATING SPARSE AND LOWDIMENSIONAL SIGNAL SEQUENCES FROM TIMEVARYING UNDERSAMPLED PROJECTIONS OF THEIR SUMS
"... The goal of this work is to recover a sequence of sparse vectors, st; and a sequence of dense vectors, ℓt, that lie in a “slowly changing” low dimensional subspace, from timevarying undersampled linear projections of their sum. This type of problem typically occurs when the quantity being imaged ca ..."
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The goal of this work is to recover a sequence of sparse vectors, st; and a sequence of dense vectors, ℓt, that lie in a “slowly changing” low dimensional subspace, from timevarying undersampled linear projections of their sum. This type of problem typically occurs when the quantity being imaged can be split into a sum of two layers, one of which is sparse and the other is lowdimensional. A key application where this problem occurs is in undersampled functional magnetic resonance imaging (fMRI) to detect brain activation patterns in response to a stimulus. The brain image at time t can be modeled as being a sum of the active region image, st, (equal to the activation in the active region and zero everywhere else) and the background brain image, ℓt, which can be accurately modeled as lying in a slowly changing low dimensional subspace. We introduce a novel solution approach called matrix completion projected compressive sensing or MatComProCS. Significantly improved performance of MatComProCS over existing work is shown for the undersampled fMRI based brain active region detection problem. Index Terms — matrix completion, compressive sensing, fMRI 1.
1 Recursive Robust PCA or Recursive Sparse Recovery in Large but Structured Noise
"... This work studies the recursive robust principal components analysis (PCA) problem. Here, “robust ” refers to robustness to both independent and correlated sparse outliers. If the outlier is the signalofinterest, this problem can be interpreted as one of recursively recovering a time sequence of s ..."
Abstract
 Add to MetaCart
This work studies the recursive robust principal components analysis (PCA) problem. Here, “robust ” refers to robustness to both independent and correlated sparse outliers. If the outlier is the signalofinterest, this problem can be interpreted as one of recursively recovering a time sequence of sparse vectors, St, in the presence of large but structured noise, Lt. The structure that we assume on Lt is that Lt is dense and lies in a low dimensional subspace that is either fixed or changes “slowly enough”. We do not assume any model on the sequence of sparse vectors. Their support sets and their nonzero element values may be either independent or correlated over time (usually in many applications they are correlated). The only thing required is that there be some support change every so often. A key application where this problem occurs is in video surveillance where the goal is to separate a slowly changing background (Lt) from moving foreground objects (St) onthefly. To solve the above problem, we introduce a novel solution called Recursive Projected CS (ReProCS). Under mild assumptions, we show that, with high probability (w.h.p.), ReProCS can exactly recover the support set of St at all times; and the reconstruction errors of both St and Lt are upper bounded by a timeinvariant and small value at all times.
Author manuscript, published in "Neural Information Processing Systems (NIPS 2012), United States (2012)" Fused sparsity and robust estimation for linear
"... models with unknown variance ..."