Provable Subspace Tracking From Missing Data and Matrix Completion

Narayanamurthy, Praneeth; Daneshpajooh, Vahid; Vaswani, Namrata

doi:10.1109/tsp.2019.2924595

Cited by 20 publications

(17 citation statements)

References 42 publications

(113 reference statements)

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“…Other subspace tracking (ST) problems that have been extensively studied include dynamic compressive sensing [58] (a special case of ST where the subspace is defined by the span of a subset of r vectors from a known dictionary matrix), dynamic robust PCA (or robust ST), see [55], [56] and references therein, streaming PCA with missing data [57], [59], and ST with missing data [60]- [64]. In terms of works with complete provable guarantees, there is the nearly optimal robust subspace tracking via recursive projected compressive sensing approach [55], [56], [64] and its precursors; recent papers on streaming PCA with missing data [57], [59], and older work on dynamic compressive sensing (CS) [58]. For robust ST, the problem setting itself implies m = n/2.…”

Section: ) Related Workmentioning

confidence: 99%

Provable Low Rank Phase Retrieval

Nayer

Narayanamurthy

Vaswani

2020

IEEE Trans. Inform. Theory

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We study the "Low Rank Phase Retrieval (LRPR)" problem defined as follows: recover an n × q matrix X * of rank r from a different and independent set of m phaseless (magnitude-only) linear projections of each of its columns. To be precise, we need to recover X * from y k := |A k x * k |, k = 1, 2, . . . , q when the measurement matrices A k are mutually independent. Here y k is an m length vector and denotes transpose. The question is when can we solve LRPR with m n? Our work introduces the first provably correct solution, Alternating Minimization for Low-Rank Phase Retrieval (AltMinLowRaP), for solving LRPR. We demonstrate its advantage over existing work via extensive simulation, and some partly real data, experiments. Our guarantee for AltMinLowRaP shows that it can solve LRPR to accuracy if mq ≥ Cnr 4 log(1/ ), the matrices A k contain i.i.d. standard Gaussian entries, the condition number of X * is bounded by a numerical constant, and its right singular vectors satisfy the incoherence (denseness) assumption from matrix completion literature. Its time complexity is only Cmqnr log 2 (1/ ). In the regime of small r, our sample complexity is much better than what standard PR methods need; and it is only about r 3 times worse than its order-optimal value of (n + q)r. Moreover, if we replace m by its lower bound for each approach, then the same can be said for the time complexity comparison with standard PR. We also briefly study the dynamic extension of LRPR.The LRPR problem occurs in phaseless dynamic imaging, e.g., Fourier ptychographic imaging of live biological specimens, where acquiring measurements is expensive. We should point out that LRPR is a very different problem than its A k = A version, or its A k = A and with-phase (linear) version, both of which have been extensively studied in the literature.for each k. If we assume κ is a constant, up to constant factors, (4) also implies (3). Thus, up to constant factors, requiring right incoherence is the same as requiring that the maximum energy of any signal x * k is within constant factors of the average.

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Section: ) Related Workmentioning

confidence: 99%

Provable Low Rank Phase Retrieval

Nayer

Narayanamurthy

Vaswani

2020

IEEE Trans. Inform. Theory

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“…Recursive projected compressive sensing (ReProCS)-based algorithms [20], [21] are also able to adaptively reconstruct a subspace from missing observations. They provide not only a memory-efficient solution, but also a precise subspace estimation as compared to the state-of-the-arts.…”

Section: A Related Workmentioning

confidence: 99%

“…Among the subspace tracking algorithms reviewed above, only a few of them are robust in the presence of both outliers and missing observations, including GRASTA [15], pROST [22], ROSETA [18], ReProCS-based algorithms [20], [21] and PETRELS-CFAR [19].…”

Section: A Related Workmentioning

confidence: 99%

“…Moreover, PETRELS-ADMM can be classified to a class of provable ST algorithms [20], [21] where a performance guarantee is provided. Our proposed algorithm takes both advantages of streaming solution (need only single-pass of data) and preserved convergence.…”

Section: B Contributionsmentioning

confidence: 99%

“…State-of-the-art algorithms for comparison are: GRASTA [15], ROSETA [18] and PETRELS-CFAR [19], ReProCS [20] and NORST [21]. Throughout our experiments, their algorithm parameters are set by default as mentioned in the algorithms.…”

Section: A Robust Subspace Trackingmentioning

confidence: 99%

See 2 more Smart Citations

Robust Subspace Tracking With Missing Data and Outliers: Novel Algorithm With Convergence Guarantee

Thanh

Dũng

Trung

et al. 2021

IEEE Trans. Signal Process.

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In this paper, we propose a novel algorithm, namely PETRELS-ADMM, to deal with subspace tracking in the presence of outliers and missing data. The proposed approach consists of two main stages: outlier rejection and subspace estimation. In the first stage, alternating direction method of multipliers (ADMM) is effectively exploited to detect outliers affecting the observed data. In the second stage, we propose an improved version of the parallel estimation and tracking by recursive least squares (PETRELS) algorithm to update the underlying subspace in the missing data context. We then present a theoretical convergence analysis of PETRELS-ADMM which shows that it generates a sequence of subspace solutions converging to the optimum of its batch counterpart. The effectiveness of the proposed algorithm, as compared to state-of-the-art algorithms, is illustrated on both simulated and real data.

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Online Tensor Robust Principal Component Analysis

Salut

Anderson

2022

IEEE Access

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Online robust principal component analysis (RPCA) algorithms recursively decompose incoming data into low-rank and sparse components. However, they operate on data vectors and cannot directly be applied to higher-order data arrays (e.g. video frames). In this paper, we propose a new online robust PCA algorithm that preserves the multi-dimensional structure of data. Our algorithm is based on the recently proposed tensor singular value decomposition (T-SVD). We develop a convex optimization-based approach to recover the sparse component; and subsequently, update the low-rank component using incremental T-SVD. We propose an efficient tensor convolutional extension to the fast iterative shrinkage thresholding algorithm (FISTA) to produce a fast algorithm to solve this optimization problem. We demonstrate tensor-RPCA with the application of background foreground separation in a video stream. The foreground component is modeled as a sparse signal. The background component is modeled as a gradually changing low-rank subspace. Extensive experiments on real-world videos are presented and results demonstrate the effectiveness of our online tensor robust PCA.INDEX TERMS Multilinear subspace learning, tensor convolutional sparse coding, low-rank tensor model, tensor singular value decomposition (T-SVD).

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Provable Subspace Tracking From Missing Data and Matrix Completion

Cited by 20 publications

References 42 publications

Provable Low Rank Phase Retrieval

Provable Low Rank Phase Retrieval

Robust Subspace Tracking With Missing Data and Outliers: Novel Algorithm With Convergence Guarantee

Online Tensor Robust Principal Component Analysis

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