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Dynamic sparse state estimation using ℓ1ℓ1 minimization: Adaptiverate measurement bounds, algorithms, and applications
 in IEEE Intern. Conf. Acoustics, Speech, and Sig. Proc. (ICASSP), 2015
"... We propose a recursive algorithm for estimating timevarying signals from a few linear measurements. The signals are assumed sparse, with unknown support, and are described by a dynamical model. In each iteration, the algorithm solves an ℓ1ℓ1 minimization problem and estimates the number of measur ..."
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We propose a recursive algorithm for estimating timevarying signals from a few linear measurements. The signals are assumed sparse, with unknown support, and are described by a dynamical model. In each iteration, the algorithm solves an ℓ1ℓ1 minimization problem and estimates the number of measurements that it has to take at the next iteration. These estimates are computed based on recent theoretical results for ℓ1ℓ1 minimization. We also provide sufficient conditions for perfect signal reconstruction at each time instant as a function of an algorithm parameter. The algorithm exhibits high performance in compressive tracking on a real video sequence, as shown in our experimental results. Index Terms — State estimation, sparsity, background subtraction, motion estimation, online algorithms
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.
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.