Robust CUR Decomposition: Theory and Imaging Applications

Cai, HanQin; Hamm, Keaton; Huang, Longxiu; Needell, Deanna

doi:10.48550/arxiv.2101.05231

Cited by 4 publications

(3 citation statements)

References 39 publications

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“…It has been used in exploratory data analyses related to natural language processing [401] and subspace clustering [402]. CUR is also used as a fast approximation to SVD [394,398,403,404], and is used to accelerate algorithms for robust PCA [405,406]. The Nyström method is the CUR decomposition where the same columns and rows are selected to approximate symmetric positive semidefinite matrices.…”

Section: Decomposition and Cur Decompositionmentioning

confidence: 99%

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Swamy

Wang

et al. 2023

Mathematics

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Compressed sensing is an alternative to Shannon/Nyquist sampling for acquiring sparse or compressible signals. Sparse coding represents a signal as a sparse linear combination of atoms, which are elementary signals derived from a predefined dictionary. Compressed sensing, sparse approximation, and dictionary learning are topics similar to sparse coding. Matrix completion is the process of recovering a data matrix from a subset of its entries, and it extends the principles of compressed sensing and sparse approximation. The nonnegative matrix factorization is a low-rank matrix factorization technique for nonnegative data. All of these low-rank matrix factorization techniques are unsupervised learning techniques, and can be used for data analysis tasks, such as dimension reduction, feature extraction, blind source separation, data compression, and knowledge discovery. In this paper, we survey a few emerging matrix factorization techniques that are receiving wide attention in machine learning, signal processing, and statistics. The treated topics are compressed sensing, dictionary learning, sparse representation, matrix completion and matrix recovery, nonnegative matrix factorization, the Nyström method, and CUR matrix decomposition in the machine learning framework. Some related topics, such as matrix factorization using metaheuristics or neurodynamics, are also introduced. A few topics are suggested for future investigation in this article.

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Section: Decomposition and Cur Decompositionmentioning

confidence: 99%

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Swamy

Wang

et al. 2023

Mathematics

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“…For example, CUR can pick any subset of r linearly independent rows and columns to obtain exact decompositions of any rank-r matrix, but it is not true for CoS-NMF. For more information on CUR decomposition and pseudo-skeleton approximation, we refer the interest reader to [4,20,26,36] and the references therein. In Section 3, we will discuss the connection and difference between CoS-NMF and CUR in details.…”

Section: Related Problemsmentioning

confidence: 99%

Co-Separable Nonnegative Matrix Factorization

Pan,

2021

Preprint

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Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. It aims to find a low rank approximation for nonnegative data M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under separability assumption, which requires the columns of factor matrix are equal to columns of the input matrix. In this paper, we generalize separability assumption based on 3-factor NMF M = P1SP2, and require that S is a sub-matrix of the input matrix. We refer to this NMF as a Co-Separable NMF (CoS-NMF). We discuss some mathematics properties of CoS-NMF, and present the relationships with other related matrix factorizations such as CUR decomposition, generalized separable NMF(GS-NMF), and bi-orthogonal trifactorization (BiOR-NM3F). An optimization model for CoS-NMF is proposed and alternated fast gradient method is employed to solve the model. Numerical experiments on synthetic datasets, document datasets and facial databases are conducted to verify the effectiveness of our CoS-NMF model. Compared to state-of-the-art methods, CoS-NMF model performs very well in co-clustering task, and preserves a good approximation to the input data matrix as well.

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“…Over the last decade, robust principal component analysis (RPCA), one of the fundamental dimension reduction techniques, has received intensive investigations from theoretical and empirical perspectives [1][2][3][4][5][6][7][8][9][10][11][12][13]. RPCA also plays a key role in a wide range of machine learning tasks, such as video background subtraction [14], singing-voice separation [15], face modeling [16], image alignment [17] , feature identification [18], community detection [19], fault detection [20], and NMR spectrum recovery [21].…”

Section: Introductionmentioning

confidence: 99%

Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection

Cai

Liu

Yin

2021

Preprint

Self Cite

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Robust principal component analysis (RPCA) is a critical tool in modern machine learning, which detects outliers in the task of low-rank matrix reconstruction. In this paper, we propose a scalable and learnable non-convex approach for highdimensional RPCA problems, which we call Learned Robust PCA (LRPCA). LRPCA is highly efficient, and its free parameters can be effectively learned to optimize via deep unfolding. Moreover, we extend deep unfolding from finite iterations to infinite iterations via a novel feedforward-recurrent-mixed neural network model. We establish the recovery guarantee of LRPCA under mild assumptions for RPCA. Numerical experiments show that LRPCA outperforms the state-of-the-art RPCA algorithms, such as ScaledGD and AltProj, on both synthetic datasets and real-world applications.

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Robust CUR Decomposition: Theory and Imaging Applications

Cited by 4 publications

References 39 publications

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Matrix Factorization Techniques in Machine Learning, Signal Processing, and Statistics

Co-Separable Nonnegative Matrix Factorization

Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection

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