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Efficient Decoupling Algorithms for Best Rank-One Tensor Approximation


Core Concepts
The author presents the Higher-Order Self-Consistent Field (HOSCF) algorithm as an efficient decoupling method for finding the best rank-one approximation of higher-order tensors, inspired by computational chemistry's self-consistent field iteration.
Abstract
The content discusses the importance of finding the best rank-one approximation of higher-order tensors and introduces the HOSCF algorithm. It explains how HOSCF is inspired by self-consistent field iteration and offers improved convergence compared to existing algorithms. The paper also covers computational complexity, stopping criteria, and an improved version of HOSCF. The convergence analysis establishes that HOSCF is locally q-linearly convergent with a defined convergence rate.
Stats
Numerical experiments show proposed algorithms efficiently converge. Proposed iHOSCF accelerates convergence speed comparable to ASVD.
Quotes

Key Insights Distilled From

by Chuanfu Xiao... at arxiv.org 03-05-2024

https://arxiv.org/pdf/2403.01778.pdf
HOSCF

Deeper Inquiries

How does the HOSCF algorithm compare to other decoupling methods in terms of efficiency

The HOSCF algorithm stands out from other decoupling methods in terms of efficiency due to its ability to update the factors simultaneously by solving the largest magnitude eigenpair of a symmetric matrix. This approach allows for high parallel efficiency, making it suitable for modern multi-core parallel computers. Unlike traditional algorithms like HOPM and ASVD, which have interdependent factor updates within each iteration leading to low parallel efficiency, HOSCF's decoupling strategy enhances scalability on parallel architectures. Additionally, the incorporation of Rayleigh quotient iteration in iHOSCF further accelerates convergence speed compared to existing methods like GRQI.

What are potential applications beyond tensor computation for the concepts introduced in this paper

The concepts introduced in this paper have potential applications beyond tensor computation in various fields such as machine learning, data analysis, and signal processing. For instance: In machine learning: The rank-one approximation problem is fundamental in dimensionality reduction techniques like Principal Component Analysis (PCA) and can be applied to reduce computational complexity. In data analysis: Tensor decomposition methods are useful for extracting patterns from multidimensional datasets such as images or videos. In signal processing: Efficient algorithms for finding best rank-one approximations can enhance noise reduction techniques or feature extraction processes.

How can the findings in this content be applied to real-world problems outside of mathematics

The findings presented in this content can be applied to real-world problems outside mathematics by leveraging the efficient decoupling algorithms developed for higher-order tensors. Some practical applications include: Image and video compression: By utilizing tensor decomposition techniques inspired by these algorithms, one can improve compression methods while preserving image quality. Biomedical imaging: Applying rank-one approximation strategies could help analyze complex medical imaging data more effectively. Natural language processing: Tensor computations can enhance text mining tasks by capturing relationships between words or phrases across multiple dimensions efficiently.
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