Operator SVD with Neural Networks via Nested Low-Rank Approximation

• URL: http://arxiv.org/abs/2402.03655v2
• Date: Wed, 21 Aug 2024 05:09:53 GMT
• Title: Operator SVD with Neural Networks via Nested Low-Rank Approximation
• Authors: J. Jon Ryu, Xiangxiang Xu, H. S. Melihcan Erol, Yuheng Bu, Lizhong Zheng, Gregory W. Wornell,
• Abstract summary: This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition. New techniques called emphnesting for learning the top-$L$ singular values and singular functions in the correct order. We demonstrate the effectiveness of the proposed framework for use cases in computational physics and machine learning.
• Score: 19.562492156734653
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Computing eigenvalue decomposition (EVD) of a given linear operator, or finding its leading eigenvalues and eigenfunctions, is a fundamental task in many machine learning and scientific computing problems. For high-dimensional eigenvalue problems, training neural networks to parameterize the eigenfunctions is considered as a promising alternative to the classical numerical linear algebra techniques. This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition, accompanied by new techniques called \emph{nesting} for learning the top-$L$ singular values and singular functions in the correct order. The proposed method promotes the desired orthogonality in the learned functions implicitly and efficiently via an unconstrained optimization formulation, which is easy to solve with off-the-shelf gradient-based optimization algorithms. We demonstrate the effectiveness of the proposed optimization framework for use cases in computational physics and machine learning.

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