Randomized Low-Rank Approximation of Continuous Parameter-Dependent Matrices

Conceitos Básicos

This work proposes randomized algorithms for efficiently computing low-rank approximations of parameter-dependent matrices, which arise in various application areas such as computational statistics and dynamical systems. The key idea is to use constant dimension reduction matrices (DRMs) instead of independent DRMs for each parameter value, leading to computationally attractive methods, especially when the parameter-dependent matrix admits an affine linear decomposition.

Resumo

This work considers the problem of computing low-rank approximations of a matrix A(t) that depends on a parameter t in a compact set D ⊂ Rd. Such parameter-dependent matrices arise in various application areas, including Gaussian process regression, time-dependent data from dynamical systems, image processing, and natural language processing.
The authors propose to extend two popular randomized algorithms, the randomized singular value decomposition (HMT method) and the generalized Nyström method, to the parameter-dependent setting. The key idea is to use constant dimension reduction matrices (DRMs) that do not depend on the parameter t, in contrast to the standard approach of using independent DRMs for each parameter value.
The use of constant DRMs leads to significant computational savings, especially when A(t) admits an affine linear decomposition with respect to t. The authors provide a probabilistic analysis for both algorithms, deriving bounds on the expected value as well as failure probabilities for the approximation error when using Gaussian random DRMs. The theoretical results and numerical experiments show that the use of constant DRMs does not impair the effectiveness of the methods, and they reliably return quasi-best low-rank approximations.

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Principais Insights Extraídos De

Randomized low-rank approximation of parameter-dependent matrices

by Daniel Kress... às arxiv.org 04-18-2024

https://arxiv.org/pdf/2302.12761.pdf

Perguntas Mais Profundas

How would the analysis and algorithms change if the goal was to minimize the worst-case error over the parameter domain D, rather than the L2 error

To minimize the worst-case error over the parameter domain D, the analysis and algorithms would need to focus on bounding the error at each point in D rather than considering the average error over the entire domain. This would involve developing bounds that hold for all points in D simultaneously, ensuring that the approximation is accurate across the entire parameter range. Algorithms would need to be designed to handle the worst-case scenario, potentially requiring more conservative approaches to approximation to guarantee accuracy at all points in D.

Can the proposed methods be extended to handle time-varying parameter domains D(t) instead of a fixed compact set D

The proposed methods can be extended to handle time-varying parameter domains D(t) by adapting the algorithms to update the low-rank approximations as the parameter t changes. This would involve recalculating the sketches and basis matrices at each time step to account for the varying parameter values. By incorporating the time dependency into the algorithms, the low-rank approximations can be continuously updated to provide accurate results as the parameter domain evolves over time.

What are the potential applications of the proposed randomized low-rank approximation techniques in other fields, such as control theory, optimization, or scientific computing

The proposed randomized low-rank approximation techniques have a wide range of potential applications in various fields. In control theory, these methods can be used for system identification, model reduction, and controller design, where efficient approximations of parameter-dependent matrices are crucial for real-time control applications. In optimization, these techniques can be applied to large-scale optimization problems with parameter dependencies, enabling faster convergence and reduced computational complexity. In scientific computing, the methods can be utilized for solving partial differential equations with parameter variations, enabling efficient simulations and analysis of complex systems. Overall, the techniques have the potential to enhance computational efficiency and accuracy in a diverse set of applications across different domains.