Estimation of Simultaneously Sparse and Low Rank Matrices

Richard, Émile; Savalle, Pierre-André; Vayatis, Nicolas

doi:10.48550/arxiv.1206.6474

Cited by 10 publications

(11 citation statements)

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“…, M t ] = L t + S t . Other recent works that also study batch algorithms for recovering a sparse S t and a low-rank L t from M t := L t + S t or from undersampled measurements include [15], [16], [17], [18], [19], [20], [21], [22], [23], [24].…”

Section: A Related Workmentioning

confidence: 99%

Recursive robust PCA or recursive sparse recovery in large but structured noise

Qiu

Vaswani

Hogben

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

169

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Abstract-This work studies the recursive robust principal components analysis (PCA) problem. If the outlier is the signalof-interest, 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." 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) on-the-fly. To solve the above problem, in recent work, we introduced a novel solution called Recursive Projected CS (ReProCS). In this work we develop a simple modification of the original ReProCS idea and analyze it. This modification assumes knowledge of a subspace change model on the Lt's. Under mild assumptions and a denseness assumption on the unestimated part of the subspace of Lt at various times, we show that, with high probability (w.h.p.), the proposed approach 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 time-invariant and small value. In simulation experiments, we observe that the last assumption holds as long as there is some support change of St every few frames.

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Section: A Related Workmentioning

confidence: 99%

Recursive robust PCA or recursive sparse recovery in large but structured noise

Qiu

Vaswani

Hogben

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

169

View full text Add to dashboard Cite

show abstract

“…, M t ] = L t + S t . Other recent works that also study batch algorithms for recovering a sparse S t and a low-rank L t from M t := L t + S t or from undersampled measurements include [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. It was shown in [5] that by solving PCP:…”

Section: Introductionmentioning

confidence: 99%

An Online Algorithm for Separating Sparse and Low-Dimensional Signal Sequences From Their Sum

Guo

Qiu²,

Vaswani³

2014

IEEE Trans. Signal Process.

131

169

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Abstract-This paper designs and extensively evaluates an online algorithm, called practical recursive projected compressive sensing (Prac-ReProCS), for recovering a time sequence of sparse vectors St and a time sequence of dense vectors Lt from their sum, Mt := St + Lt, when the Lt's lie in a slowly changing lowdimensional subspace of the full space. A key application where this problem occurs is in real-time 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 on-the-fly. Prac-ReProCS is a practical modification of its theoretical counterpart which was analyzed in our recent work. Experimental comparisons demonstrating the advantage of the approach for both simulated and real videos, over existing batch and recursive methods, are shown. Extension to the undersampled case is also developed.

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“…This issue is partially offset by sparse storage of the adjacency matrix, in general, and largely ameliorated by data that is on the same discretization, as in this work. In future work we will investigate low-rank approximations to the adjacency matrix [80][81][82][83][84], dimensionality reduction techniques [85,86], and the use of graph auto-encoders [87][88][89][90] to reduce the mesh-based graphs in-line. We are also pursuing the larger topic of processing image with multi-resolution filters [91], e.g.…”

Section: Discussionmentioning

confidence: 99%

Mesh-based graph convolutional neural networks for modeling materials with microstructure

Frankel¹,

Safta²,

Alleman³

et al. 2021

Preprint

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Predicting the evolution of a representative sample of a material with microstructure is a fundamental problem in homogenization. In this work we propose a graph convolutional neural network that utilizes the discretized representation of the initial microstructure directly, without segmentation or clustering. Compared to feature-based and pixel-based convolutional neural network models, the proposed method has a number of advantages: (a) it is deep in that it does not require featurization but can benefit from it, (b) it has a simple implementation with standard convolutional filters and layers, (c) it works natively on unstructured and structured grid data without interpolation (unlike pixel-based convolutional neural networks), and (d) it preserves rotational invariance like other graph-based convolutional neural networks. We demonstrate the performance of the proposed network and compare it to traditional pixel-based convolution neural network models and feature-based graph convolutional neural networks on three large datasets.

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Estimation of Simultaneously Sparse and Low Rank Matrices

Cited by 10 publications

References 0 publications

Recursive robust PCA or recursive sparse recovery in large but structured noise

Recursive robust PCA or recursive sparse recovery in large but structured noise

An Online Algorithm for Separating Sparse and Low-Dimensional Signal Sequences From Their Sum

Mesh-based graph convolutional neural networks for modeling materials with microstructure

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