Decentralized Rank-Adaptive Matrix Factorization-Part I: Algorithm Development

Factorizing a low-rank matrix into two matrix factors with low dimensions from its noisy observations is a classical but challenging problem arising from real-world applications. This paper develops decentralized matrix factorization algorithms, i.e., factorizing a matrix whose columns are stored di...

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Vydáno v:	IEEE transactions on signal processing Ročník 73; s. 4124 - 4140
Hlavní autoři:	Jiao, Yuchen, Gu, Yuantao, Chang, Tsung-Hui, Luo, Zhi-Quan
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York IEEE 2025 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Accuracy Adaptive algorithms Algorithms Convergence decentralized Factorization Image reconstruction Linear programming Low rank Matrix decomposition matrix factorization Noise Principal component analysis rank adaptive Regularization Reviews Signal processing algorithms Sparse matrices
ISSN:	1053-587X, 1941-0476
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Abstract	Factorizing a low-rank matrix into two matrix factors with low dimensions from its noisy observations is a classical but challenging problem arising from real-world applications. This paper develops decentralized matrix factorization algorithms, i.e., factorizing a matrix whose columns are stored distributively over a network without a central agent. The performance of existing algorithms relies heavily on the accuracy of the matrix rank estimate. However, acquiring an accurate estimate is difficult in the distributed setting. In this paper and its Part II, we address this problem by introducing a novel regularization into the objective function to induce the solution with correct rank. Based on this, we propose a rank-adaptive decentralized MF algorithm. In Part I, we delineate the algorithm development from the centralized with known rank, decentralized with known rank, to the rank-adaptive decentralized settings. For the centralized algorithm, we present the first globally linear convergence analysis for the alternating gradient descent method. In the Part II, we analyze conditions for which the proposed rank-adaptive decentralized MF algorithm converges to the global solution with the correct rank. Numerical experiments based on both synthetic and real-world datasets are presented in this paper to demonstrate the effectiveness of the proposed algorithms and corroborate the theoretical claims.
AbstractList	Factorizing a low-rank matrix into two matrix factors with low dimensions from its noisy observations is a classical but challenging problem arising from real-world applications. This paper develops decentralized matrix factorization algorithms, i.e., factorizing a matrix whose columns are stored distributively over a network without a central agent. The performance of existing algorithms relies heavily on the accuracy of the matrix rank estimate. However, acquiring an accurate estimate is difficult in the distributed setting. In this paper and its Part II, we address this problem by introducing a novel regularization into the objective function to induce the solution with correct rank. Based on this, we propose a rank-adaptive decentralized MF algorithm. In Part I, we delineate the algorithm development from the centralized with known rank, decentralized with known rank, to the rank-adaptive decentralized settings. For the centralized algorithm, we present the first globally linear convergence analysis for the alternating gradient descent method. In the Part II, we analyze conditions for which the proposed rank-adaptive decentralized MF algorithm converges to the global solution with the correct rank. Numerical experiments based on both synthetic and real-world datasets are presented in this paper to demonstrate the effectiveness of the proposed algorithms and corroborate the theoretical claims.
Author	Luo, Zhi-Quan Chang, Tsung-Hui Jiao, Yuchen Gu, Yuantao
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Snippet	Factorizing a low-rank matrix into two matrix factors with low dimensions from its noisy observations is a classical but challenging problem arising from...
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SubjectTerms	Accuracy Adaptive algorithms Algorithms Convergence decentralized Factorization Image reconstruction Linear programming Low rank Matrix decomposition matrix factorization Noise Principal component analysis rank adaptive Regularization Reviews Signal processing algorithms Sparse matrices
Title	Decentralized Rank-Adaptive Matrix Factorization-Part I: Algorithm Development
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