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...(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 low-rank or low-rank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refer...

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...(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 low-rank or low-rank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refer...

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...(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 low-rank or low-rank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refer...

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...(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 low-rank or low-rank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refer...

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... user rates only a few movies. Low-rank plus sparse matrix recovery refers to recovering a sparse matrix and a low-rank matrix from their sum or from a subset of entries of their sum. As explained in =-=[7]-=-, this can equivalently be understood as a robust principal components’ analysis (PCA) problem. An important application where it occurs is video layering (separating a video sequence into foreground ...

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...(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 low-rank or low-rank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refer...

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... heuristics have been proposed for it, e.g., see [10] and references therein. However none of the practically useful algorithms from older literature come with performance guarantees. In recent works =-=[7, 11]-=-, Candès et. al. and Chandrasekaran et. al. posed robust PCA as a problem of separating a sparse matrix S and a low-rank matrix L from their sum, Y := L+ S. They introduced the Principal Components P...

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... magnitude. Robust PCA, which refers to the problem of PCA in the presence of outliers, has been a well-studied problem for a long time and many useful heuristics have been proposed for it, e.g., see =-=[10]-=- and references therein. However none of the practically useful algorithms from older literature come with performance guarantees. In recent works [7, 11], Candès et. al. and Chandrasekaran et. al. p...

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...-PCP [24, 25]. Modified-PCP solves the problem of robust PCA with partial subspace knowledge using an idea inspired by our older work on modified-CS for sparse recovery with partial support knowledge =-=[26]-=- (explained below). The advantage 3 of modified-PCP is that it needs a weaker assumption on the rank-sparsity product compared to both PCP and ReProCS. However, its disadvantage is similar to that of ...

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...in, heart, larynx or other human organ images are piecewise smooth, and thus approximately sparse in the wavelet domain. In a time sequence, their sparsity pattern changes with time, but quite slowly =-=[27, 28, 29]-=-. This simple idea, which was first used in [27], is the key reason our proposed algorithms can achieve accurate reconstruction from much fewer measurements. In recent years, the static sparse recover...

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...easurements. In recent years, the static sparse recovery or compressive sensing (CS) problem has been thoroughly studied [1, 2, 3, 4, 5, 6]. But most existing algorithms for the dynamic problem, e.g. =-=[30, 31]-=-, just use CS solutions to jointly reconstruct the entire time sequence in one go. This is a batch solution and as a result (a) it is very slow in recovering a long sequence and (b) its memory require...

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...(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 low-rank or low-rank plus sparse matrix recovery, have also been studied in detail. Sparse recovery or CS refer...

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...r batch robust PCA approaches. Moreover, almost all existing work on finite sample PCA assumes that the error between the measured and true data vectors is uncorrelated with the true data, see, e.g., =-=[23]-=- and references therein. However, in case of ReProCS, because of how the estimate ˆ̀t is computed, the error et := ˆ̀t − `t is correlated with `t and so our proof could not just combine a result for s...

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...from their sum, Y := L+ S. They introduced the Principal Components Pursuit (PCP) program and obtained performance guarantees for it under mostly mild assumptions [7], [11]. Later work by Hsu et. al. =-=[12]-=- improved the result of [11]. Since then, there has been much later work on provably accurate robust PCA solutions but all of it has been for batch methods. In our work, we consider an online or recur...

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... it is highly sensitive to outliers. “Outlier” is a loosely defined term that refers to any corruption that is not small compared to the true data vector and that occurs occasionally. As suggested in =-=[9]-=-, an outlier can be nicely modeled as a sparse vector whose nonzero entries can have any magnitude. Robust PCA, which refers to the problem of PCA in the presence of outliers, has been a well-studied ...

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..., ReProCS significantly outperforms batch robust PCA approaches, such as PCP and RSL [10], as well as online algorithms such as incremental RSL [19], and a recently introduced algorithm called GRASTA =-=[20]-=-. The experimental comparisons are available in [16, 21] and at http://www.ece.iastate.edu/˜hanguo/PracReProCS.html. [Correctness result for ReProCS] In recent work [17, 18], we have shown that, as lo...

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...Vector (MMV) problem, but this assumes that the sparsity pattern of the signal sequence is constant with time. This assumption is often not valid either. To the best of our knowledge, our recent work =-=[27, 28, 32, 33]-=- proposed the first solutions for recursively reconstructing sparse signal sequences using much fewer measurements than those needed for accurate recovery using simple-CS methods. The computational an...

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...as a short initial background-only sequence is available, ReProCS significantly outperforms batch robust PCA approaches, such as PCP and RSL [10], as well as online algorithms such as incremental RSL =-=[19]-=-, and a recently introduced algorithm called GRASTA [20]. The experimental comparisons are available in [16, 21] and at http://www.ece.iastate.edu/˜hanguo/PracReProCS.html. [Correctness result for ReP...

Citation Context

...easurements. In recent years, the static sparse recovery or compressive sensing (CS) problem has been thoroughly studied [1, 2, 3, 4, 5, 6]. But most existing algorithms for the dynamic problem, e.g. =-=[30, 31]-=-, just use CS solutions to jointly reconstruct the entire time sequence in one go. This is a batch solution and as a result (a) it is very slow in recovering a long sequence and (b) its memory require...

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...us behavior in dynamic social networks; or recommendation system design in the presence of outliers and missing data.In most of these applications, an online solution is desirable. In our recent work =-=[13, 14, 15, 16, 17, 18]-=-, we have introduced a novel online solution called Recursive Projected CS (ReProCS); obtained a correctness result for it under mild assumptions; and shown that it significantly outperforms most exis...

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...us behavior in dynamic social networks; or recommendation system design in the presence of outliers and missing data.In most of these applications, an online solution is desirable. In our recent work =-=[13, 14, 15, 16, 17, 18]-=-, we have introduced a novel online solution called Recursive Projected CS (ReProCS); obtained a correctness result for it under mild assumptions; and shown that it significantly outperforms most exis...

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...ce guarantee for it. However the result was not a correctness result (it depended on intermediate algorithm estimates satisfying a certain property). The same is true for a later result of Feng et al =-=[22]-=-. 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 provid...

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...in, heart, larynx or other human organ images are piecewise smooth, and thus approximately sparse in the wavelet domain. In a time sequence, their sparsity pattern changes with time, but quite slowly =-=[27, 28, 29]-=-. This simple idea, which was first used in [27], is the key reason our proposed algorithms can achieve accurate reconstruction from much fewer measurements. In recent years, the static sparse recover...

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...us behavior in dynamic social networks; or recommendation system design in the presence of outliers and missing data.In most of these applications, an online solution is desirable. In our recent work =-=[13, 14, 15, 16, 17, 18]-=-, we have introduced a novel online solution called Recursive Projected CS (ReProCS); obtained a correctness result for it under mild assumptions; and shown that it significantly outperforms most exis...

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...taining similar error bounds on simple-CS error, both the support recovery errors and the reconstruction errors, are “stable”, i.e. they remain bounded by time-invariant and small values at all times =-=[32, 37, 38, 39]-=-. Stability is critical for any recursive algorithm since it ensures that the reconstruction error does not blow up over time. In practice, say in real-time MRI, provably stable and small error would ...

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...l subspace assumption on a single vector). The resulting solution can be understood as a modification of the PCP program when partial subspace knowledge is available and hence we call it modified-PCP =-=[24, 25]-=-. Modified-PCP solves the problem of robust PCA with partial subspace knowledge using an idea inspired by our older work on modified-CS for sparse recovery with partial support knowledge [26] (explain...

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... approaches, such as PCP and RSL [10], as well as online algorithms such as incremental RSL [19], and a recently introduced algorithm called GRASTA [20]. The experimental comparisons are available in =-=[16, 21]-=- and at http://www.ece.iastate.edu/˜hanguo/PracReProCS.html. [Correctness result for ReProCS] In recent work [17, 18], we have shown that, as long as the ReProCS algorithm parameters are set appropria...

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...s also change gradually over time [34]. The first assumption above is a new assumption introduced in our work [27, 28]. It has been empirically verified both for medical image sequences and for video =-=[32, 33, 35]-=-. The second assumption is commonly used both for adaptive filtering as well as for various tracking problems. 4 When using only assumption 1 above, the above problem can be reformulated as one of spa...

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...construction from very few measurements [33]. By also using slow signal value change, one can design regularized modified-CS which also constrains the change of the nonzero coefficient values along T =-=[34, 36]-=-. In numerical experiments as well as in experiments with simulated dynamic MR imaging, modified-CS significantly outperformed existing work at the time [33]. Under the practically valid assumption of...

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...us behavior in dynamic social networks; or recommendation system design in the presence of outliers and missing data.In most of these applications, an online solution is desirable. In our recent work =-=[13, 14, 15, 16, 17, 18]-=-, we have introduced a novel online solution called Recursive Projected CS (ReProCS); obtained a correctness result for it under mild assumptions; and shown that it significantly outperforms most exis...

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...taining similar error bounds on simple-CS error, both the support recovery errors and the reconstruction errors, are “stable”, i.e. they remain bounded by time-invariant and small values at all times =-=[32, 37, 38, 39]-=-. Stability is critical for any recursive algorithm since it ensures that the reconstruction error does not blow up over time. In practice, say in real-time MRI, provably stable and small error would ...

Citation Context

...us behavior in dynamic social networks; or recommendation system design in the presence of outliers and missing data.In most of these applications, an online solution is desirable. In our recent work =-=[13, 14, 15, 16, 17, 18]-=-, we have introduced a novel online solution called Recursive Projected CS (ReProCS); obtained a correctness result for it under mild assumptions; and shown that it significantly outperforms most exis...

Citation Context

...l subspace assumption on a single vector). The resulting solution can be understood as a modification of the PCP program when partial subspace knowledge is available and hence we call it modified-PCP =-=[24, 25]-=-. Modified-PCP solves the problem of robust PCA with partial subspace knowledge using an idea inspired by our older work on modified-CS for sparse recovery with partial support knowledge [26] (explain...