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Recursive robust pca or recursive sparse recovery in large but structured noise
 in IEEE Intl. Symp. on Information Theory (ISIT
, 2013
"... This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more informati ..."
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Cited by 22 (17 self)
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This Dissertation is brought to you for free and open access by the Graduate College at Digital Repository @ Iowa State University. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Digital Repository @ Iowa State University. For more information, please contact
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.
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.
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 ..."
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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.
1Robust PCA with Partial Subspace Knowledge
"... Abstract—In 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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Abstract—In 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. I.
Namrata Vaswani Online Robust PCA 1/54 Online Sparse Matrix Recovery Online Robust PCA
"... Recovery from incomplete data: the question I In many applications, data acquisition is slow, e.g. in MRI, acquire one Fourier coefficient of the crosssection of interest at a time I Question: can we recover the crosssection’s image from undersampled data? I Yes: if spatiallylimited or if exploit ..."
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Recovery from incomplete data: the question I In many applications, data acquisition is slow, e.g. in MRI, acquire one Fourier coefficient of the crosssection of interest at a time I Question: can we recover the crosssection’s image from undersampled data? I Yes: if spatiallylimited or if exploit sparsity of the image in an appropriate domain I In many other applications, data acquisition is fast but cannot see everything, e.g. in video, image = background + foreground I Question: can we recover two image sequences from one? I Yes: if exploit the lowrank structure of the background sequence and sparseness of the foreground
Research Statement
"... My research lies at the intersection of machine learning for high dimensional problems, signal and information processing and applications in video, bigdata analytics and bioimaging. More specifically, I have worked on designing and analyzing online algorithms for various highdimensional structur ..."
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My research lies at the intersection of machine learning for high dimensional problems, signal and information processing and applications in video, bigdata analytics and bioimaging. More specifically, I have worked on designing and analyzing online algorithms for various highdimensional structured data recovery problems and on demonstrating their usefulness in dynamic magnetic resonance imaging (MRI) and in video analytics. In the last two decades, the sparse recovery problem, or what is now more commonly referred to as compressive sensing (CS), has been extensively studied, see for example [1, 2, 3, 4, 5, 6] and later works. More recently various other structured data recovery problems, such as lowrank or lowrank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refers to the problem of recovering a sparse signal from a highly reduced set of its projected measurements. Many medical imaging techniques image crosssections of human organs noninvasively by acquiring their linear projections one at a time and then reconstructing the image from these projections. For example, in magnetic resonance imaging (MRI), one acquires Fourier projections one at a time, while in Computed Tomography (CT), one acquires the Radon transform coefficients one a time. For all these applications, the ability to accurately reconstruct using fewer measurements directly translates into reduced scan times and hence sparse recovery methods have had a huge impact in these areas. Lowrank
1A Correctness Result for Online Robust PCA Brian Lois, Graduate Student Member, IEEE
"... This work studies the problem of sequentially recovering a sparse vector xt and a vector from a lowdimensional subspace `t from knowledge of their sum mt = xt + `t. If the primary goal is to recover the lowdimensional subspace where the `t’s lie, then the problem is one of online or recursive robu ..."
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This work studies the problem of sequentially recovering a sparse vector xt and a vector from a lowdimensional subspace `t from knowledge of their sum mt = xt + `t. If the primary goal is to recover the lowdimensional subspace where the `t’s lie, then the problem is one of online or recursive robust principal components analysis (PCA). To the best of our knowledge, this is the first correctness result for online robust PCA. We prove that if the `t’s obey certain denseness and slow subspace change assumptions, and the support of xt changes by at least a certain amount at least every so often, and some other mild assumptions hold, then with high probability, the support of xt will be recovered exactly, and the error made in estimating xt and `t will be small. An example of where such a problem might arise is in separating a sparse foreground and slowly changing dense background in a surveillance video. I.
1AdaptiveRate Reconstruction of TimeVarying Signals with Application in Compressive Foreground Extraction
"... Abstract—We propose and analyze an online algorithm for reconstructing a sequence of signals from a limited number of linear measurements. The signals are assumed sparse, with unknown support, and evolve over time according to a generic nonlinear dynamical model. Our algorithm, based on recent theor ..."
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Abstract—We propose and analyze an online algorithm for reconstructing a sequence of signals from a limited number of linear measurements. The signals are assumed sparse, with unknown support, and evolve over time according to a generic nonlinear dynamical model. Our algorithm, based on recent theoretical results for ℓ1ℓ1 minimization, is recursive and computes the number of measurements to be taken at each time onthefly. As an example, we apply the algorithm to online compressive video background subtraction, a problem stated as follows: given a set of measurements of a sequence of images with a static background, simultaneously reconstruct each image while separating its foreground from the background. The performance of our method is illustrated on sequences of real images. We observe that it allows a dramatic reduction in the number of measurements or reconstruction error with respect to stateoftheart compressive background subtraction schemes. Index Terms—State estimation, compressive video, background subtraction, sparsity, ℓ1 minimization, motion estimation. I.
Recursive Recovery of Sparse Signal Sequences from Compressive Measurements: A Review
"... In this article, we review the literature on the design and analysis of recursive algorithms for reconstructing a time sequence of sparse signals from compressive measurements. The signals are assumed to be sparse in some transform domain or in some dictionary. Their sparsity patterns can change wi ..."
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In this article, we review the literature on the design and analysis of recursive algorithms for reconstructing a time sequence of sparse signals from compressive measurements. The signals are assumed to be sparse in some transform domain or in some dictionary. Their sparsity patterns can change with time, although in many practical applications, the changes are gradual. An important class of applications where this problem occurs is dynamic projection imaging, e.g., dynamic magnetic resonance imaging for realtime medical applications such as interventional radiology, or dynamic computed tomography.