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22
Online robust pca via stochastic optimization
 in Adv. Neural Info. Proc. Sys. (NIPS
, 2013
"... Robust PCA methods are typically based on batch optimization and have to load all the samples into memory during optimization. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust PCA (ORPCA) that processes one sample per time instance and hence its m ..."
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Cited by 20 (2 self)
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Robust PCA methods are typically based on batch optimization and have to load all the samples into memory during optimization. This prevents them from efficiently processing big data. In this paper, we develop an Online Robust PCA (ORPCA) that processes one sample per time instance and hence its memory cost is independent of the number of samples, significantly enhancing the computation and storage efficiency. The proposed ORPCA is based on stochastic optimization of an equivalent reformulation of the batch RPCA. Indeed, we show that ORPCA provides a sequence of subspace estimations converging to the optimum of its batch counterpart and hence is provably robust to sparse corruption. Moreover, ORPCA can naturally be applied for tracking dynamic subspace. Comprehensive simulations on subspace recovering and tracking demonstrate the robustness and efficiency advantages of the ORPCA over online PCA and batch RPCA methods. 1
An online algorithm for separating sparse and lowdimensional signal sequences from their sum
 IEEE Trans. Signal Process
"... Abstract—This paper designs and extensively evaluates an online algorithm, called practical recursive projected compressive sensing (PracReProCS), for recovering a time sequence of sparse vectors and a time sequence of dense vectors from their sum, , when the ’s lie in a slowly changing lowdimens ..."
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Abstract—This paper designs and extensively evaluates an online algorithm, called practical recursive projected compressive sensing (PracReProCS), for recovering a time sequence of sparse vectors and a time sequence of dense vectors from their sum, , when the ’s lie in a slowly changing lowdimensional subspace of the full space. A key application where this problem occurs is in realtime video layering where the goal is to separate a video sequence into a slowly changing background sequence and a sparse foreground sequence that consists of one or more moving regions/objects onthefly. PracReProCS is a practical modification of its theoretical counterpart which was analyzed in our recent work. Extension to the undersampled case is also developed. Extensive experimental comparisons demonstrating the advantage of the approach for both simulated and real videos, over existing batch and recursive methods, are shown. Index Terms—Online robust PCA, recursive sparse recovery, large but structured noise, compressed sensing. I.
Online pca for contaminated data
 in Proc. Adv. Neural Inf. Process. Syst. (NIPS
"... We consider the online Principal Component Analysis (PCA) where contaminated samples (containing outliers) are revealed sequentially to the Principal Components (PCs) estimator. Due to their sensitiveness to outliers, previous online PCA algorithms fail in this case and their results can be arbitra ..."
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Cited by 8 (0 self)
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We consider the online Principal Component Analysis (PCA) where contaminated samples (containing outliers) are revealed sequentially to the Principal Components (PCs) estimator. Due to their sensitiveness to outliers, previous online PCA algorithms fail in this case and their results can be arbitrarily skewed by the outliers. Here we propose the online robust PCA algorithm, which is able to improve the PCs estimation upon an initial one steadily, even when faced with a constant fraction of outliers. We show that the final result of the proposed online RPCA has an acceptable degradation from the optimum. Actually, under mild conditions, online RPCA achieves the maximal robustness with a 50 % breakdown point. Moreover, online RPCA is shown to be efficient for both storage and computation, since it need not reexplore the previous samples as in traditional robust PCA algorithms. This endows online RPCA with scalability for large scale data. 1
Performance guarantees for undersampled recursive sparse recovery in large but structured noise (long version).” [Online]. Available: http://www.public.iastate. edu/%7Eblois/ReProModCSLong.pdf
"... Abstract—We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt +BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, ..."
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Cited by 8 (7 self)
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Abstract—We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt +BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, and has support which is correlated over time. We introduce a solution which we call Recursive Projected Modified Compressed Sensing (ReProMoCS), which exploits the correlated support change of St. We show that, under weaker assumptions than previous work, with high probability, ReProMoCS will exactly recover the support set of St and the reconstruction error of St is upper bounded by a small timeinvariant value. A motivating application where the above problem occurs is in functional MRI imaging of the brain to detect regions that are “activated ” in response to stimuli. In this case both measurement matrices are the same (i.e. A = B). The active region image constitutes the sparse vector St and this region changes slowly over time. The background brain image changes are global but the amount of change is very little and hence it can be well modeled as lying in a slowly changing low dimensional subspace, i.e. this constitutes Lt. I.
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 ..."
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Cited by 7 (7 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.
Robust pca with partial subspace knowledge,”
 in IEEE Intl. Symp. on Information Theory (ISIT),
, 2014
"... AbstractIn recent work, robust Principal Components Analysis (PCA) has been posed as a problem of recovering a lowrank matrix L and a sparse matrix S from their sum, M := L + S and a provably exact convex optimization solution called PCP has been proposed. This work studies the following problem. ..."
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Cited by 5 (2 self)
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AbstractIn recent work, robust Principal Components Analysis (PCA) has been posed as a problem of recovering a lowrank matrix L and a sparse matrix S from their sum, M := L + S and a provably exact convex optimization solution called PCP has been proposed. This work studies the following problem. Suppose that we have partial knowledge about the column space of the low rank matrix L. Can we use this information to improve the PCP solution, i.e. allow recovery under weaker assumptions? We propose here a simple but useful modification of the PCP idea, called modifiedPCP, that allows us to use this knowledge. We derive its correctness result which shows that, when the available subspace knowledge is accurate, modifiedPCP indeed requires significantly weaker incoherence assumptions than PCP. Extensive simulations are also used to illustrate this. Comparisons with PCP and other existing work are shown for a stylized real application as well. Finally, we explain how this problem naturally occurs in many applications involving time series data, i.e. in what is called the online or recursive robust PCA problem. A corollary for this case is also given.
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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Cited by 4 (3 self)
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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.
PRACTICAL REPROCS FOR SEPARATING SPARSE AND LOWDIMENSIONAL SIGNAL SEQUENCES FROM THEIR SUM – PART 1
"... This paper designs and evaluates a practical algorithm, called PracReProCS, for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt: = St + Lt, when any subsequence of the Lt’s lies in a slowly changing lowdimensional subspace. A key applica ..."
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Cited by 3 (3 self)
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This paper designs and evaluates a practical algorithm, called PracReProCS, for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt: = St + Lt, when any subsequence of the Lt’s lies in a slowly changing lowdimensional subspace. A key application where this problem occurs is in video layering where the goal is to separate a video sequence into a slowly changing background sequence and a sparse foreground sequence that consists of one or more moving regions/objects. PracReProCS is the practical analog of its theoretical counterpart that was studied in our recent work. Index Terms — robust PCA, robust matrix completion, sparse recovery, compressed sensing 1.
Recursive projected modified compressed sensing for undersampled measurements,” http://www.public.iastate.edu/ blois/ReProModCSLong.pdf
"... Abstract—We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt + BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, ..."
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Cited by 1 (1 self)
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Abstract—We study the problem of recursively reconstructing a time sequence of sparse vectors St from measurements of the form Mt = ASt + BLt where A and B are known measurement matrices, and Lt lies in a slowly changing low dimensional subspace. We assume that the signal of interest (St) is sparse, and has support which is correlated over time. We introduce a solution which we call Recursive Projected Modified Compressed Sensing (ReProMoCS), which exploits the correlated support change of St. We show that, under weaker assumptions than previous work, with high probability, ReProMoCS will exactly recover the support set of St and the reconstruction error of St is upper bounded by a small timeinvariant value. A motivating application where the above problem occurs is in functional MRI imaging of the brain to detect regions that are “activated ” in response to stimuli. In this case both measurement matrices are the same (i.e. A = B). The active region image constitutes the sparse vector St and this region changes slowly over time. The background brain image changes are global but the amount of change is very little and hence it can be well modeled as lying in a slowly changing low dimensional subspace, i.e. this constitutes Lt. I.