Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery

Cai, HanQin; Cai, Jian-Feng; Wang, Tianming; Yin, Guojian

doi:10.1109/tsp.2021.3049618

Cited by 26 publications

(13 citation statements)

References 35 publications

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“…Remark 1. For a successful reconstruction, we expect µ i (L) = O (1), i = 1, 2, which is generally true in real-world RPCA applications [8,10,11]. For the purpose of theoretical analysis, if µ i (L) is too large, then the RPCA problem may become too difficult to solve since the toleration of α usually depends on poly( 1 µ i (L) ) [11,13,35,49].…”

Section: Preliminaries and Layoutmentioning

confidence: 91%

Robust CUR Decomposition: Theory and Imaging Applications

Cai¹,

Hamm²,

Huang³

et al. 2021

Preprint

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This paper considers the use of Robust PCA in a CUR decomposition framework and applications thereof. Our main algorithms produce a robust version of columnrow factorizations of matrices D = L + S where L is low-rank and S contains sparse outliers. These methods yield interpretable factorizations at low computational cost, and provide new CUR decompositions that are robust to sparse outliers, in contrast to previous methods. We consider two key imaging applications of Robust PCA: video foreground-background separation and face modeling. This paper examines the qualitative behavior of our Robust CUR decompositions on the benchmark videos and face datasets, and find that our method works as well as standard Robust PCA while being significantly faster. Additionally, we consider hybrid randomized and deterministic sampling methods which produce a compact CUR decomposition of a given matrix, and apply this to video sequences to produce canonical frames thereof.

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Section: Preliminaries and Layoutmentioning

confidence: 91%

Robust CUR Decomposition: Theory and Imaging Applications

Cai¹,

Hamm²,

Huang³

et al. 2021

Preprint

Self Cite

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“…However, in addition, recall that we also need to enforce the spikiness (or incoherent) condition on the low-rank estimate. While in some particular applications (Cai et al, 2020b) or through sophisticated analysis (Cai et al, 2019a), it is possible to directly prove the incoherence property of T l+1 , this approach is, if not impossible, unavailable in our general framework. Instead, we first truncate W l entry-wisely by ζ l+1 /2 for some carefully chosen threshold ζ l+1 and obtain W l .…”

Section: Riemannian Gradient Descent Algorithmmentioning

confidence: 99%

Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization

Cai¹,

Li²,

Dong³

2021

Preprint

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We investigate a generalized framework to estimate a latent low-rank plus sparse tensor, where the low-rank tensor often captures the multi-way principal components and the sparse tensor accounts for potential model mis-specifications or heterogeneous signals that are unexplainable by the low-rank part. The framework is flexible covering both linear and non-linear models, and can easily handle continuous or categorical variables. We propose a fast algorithm by integrating the Riemannian gradient descent and a novel gradient pruning procedure. Under suitable conditions, the algorithm converges linearly and can simultaneously estimate both the low-rank and sparse tensors. The statistical error bounds of final estimates are established in terms of the gradient of loss function. The error bounds are generally sharp under specific statistical models, e.g., the robust tensor PCA and the community detection in hypergraph networks with outlier vertices. Moreover, our method achieves non-trivial error bounds for heavy-tailed tensor PCA whenever the noise has a finite 2 + ε moment. We apply our method to analyze the international trade flow dataset and the statistician hypergraph co-authorship network, both yielding new and interesting findings.

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“…where rank(•) measures the matrix rank, • 0 denotes l 0 norm used to count the number of nonzero entries in the matrix, and γ > 0 is a trade-off parameter. In the literature, problem (1) can be solved by either convex relaxations-approximating the matrix rank by nuclear norm and the l 0 norm by l 1 norm under certain conditions [7,8], or nonconvex methods, such as matrix factorization [5,9] and alternating minimization [10,11]. More references about RPCA solution can be found in [12].…”

Section: Introductionmentioning

confidence: 99%

Hankel-structured Tensor Robust PCA for Multivariate Traffic Time Series Anomaly Detection

Wang,

Miranda-Moreno,

Sun

2021

Preprint

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Spatiotemporal traffic data (e.g., link speed/flow) collected from sensor networks can be organized as multivariate time series with additional spatial attributes. A crucial task in analyzing such data is to identify and detect anomalous observations and events from the data with complex spatial and temporal dependencies. Robust Principal Component Analysis (RPCA) is a widely used tool for anomaly detection. However, the traditional RPCA purely relies on the global low-rank assumption while ignoring the local temporal correlations. In light of this, this study proposes a Hankel-structured tensor version of RPCA for anomaly detection in spatiotemporal data. We treat the raw data with anomalies as a multivariate time series matrix (location × time) and assume the denoised matrix has a lowrank structure. Then we transform the low-rank matrix to a third-order tensor by applying temporal Hankelization. In the end, we decompose the corrupted matrix into a low-rank Hankel tensor and a sparse matrix. With the Hankelization operation, the model can simultaneously capture the global and local spatiotemporal correlations and exhibit more robust performance. We formulate the problem as an optimization problem and use tensor nuclear norm (TNN) to approximate the tensor rank and l1 norm to approximate the sparsity. We develop an efficient solution algorithm based on the Alternating Direction Method of Multipliers (ADMM). Despite having three hyper-parameters, the model is easy to set in practice. We evaluate the proposed method by synthetic data and metro passenger flow time series and the results demonstrate the accuracy of anomaly detection.

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Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery

Cited by 26 publications

References 35 publications

Robust CUR Decomposition: Theory and Imaging Applications

Robust CUR Decomposition: Theory and Imaging Applications

Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization

Hankel-structured Tensor Robust PCA for Multivariate Traffic Time Series Anomaly Detection

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