Core Concepts

This paper presents an infinite-dimensional extension of the randomized Nyström approximation to compute low-rank approximations of non-negative self-adjoint trace-class operators. The analysis yields bounds on the expected value and tail bounds for the Nyström approximation error in the operator, trace, and Hilbert-Schmidt norms.

Abstract

The paper introduces an infinite-dimensional analog of the randomized Nyström approximation to compute low-rank approximations of non-negative self-adjoint trace-class operators. The key contributions are:
Analysis of the finite-dimensional Nyström approximation with correlated Gaussian sketches: The authors derive expectation and probability bounds on the approximation error in the Frobenius, spectral, and nuclear norms. This generalizes existing results that assume standard Gaussian sketches.
Infinite-dimensional extension of the Nyström approximation: The authors present an infinite-dimensional analog of the Nyström approximation for Hilbert-Schmidt operators and provide an analysis of the approximation error.
Improved analysis of the infinite-dimensional randomized SVD: As a byproduct, the authors improve the existing bounds for the infinite-dimensional randomized SVD, making them match the finite-dimensional results.
The analysis relies on properties of Gaussian processes and Hilbert-Schmidt operators. Numerical experiments for simple integral operators validate the proposed framework.

Stats

The paper does not contain any explicit numerical data or statistics to extract.

Quotes

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Deeper Inquiries

The infinite-dimensional Nyström approximation has various potential applications beyond those mentioned in the paper. One such application is in the field of machine learning, particularly in kernel methods and Gaussian processes. By leveraging the Nyström approximation in infinite dimensions, it can be utilized for tasks such as kernelized regression, kernel PCA, and spectral clustering. Additionally, in the realm of computational physics and quantum mechanics, the Nyström approximation can be employed for solving integral equations and studying quantum systems with continuous spectra. Furthermore, in signal processing and image analysis, the Nyström approximation can aid in denoising, compression, and feature extraction tasks on continuous data representations.

To extend the analysis to handle non-symmetric or non-positive definite operators, modifications and adaptations would be necessary. For non-symmetric operators, the eigenvalue decomposition would need to be adjusted to accommodate the non-symmetric nature of the operator. This could involve utilizing generalized eigenvectors and eigenvalues. For non-positive definite operators, the analysis would need to consider the spectral properties of such operators, potentially involving different norms and decompositions that are suitable for non-positive definite matrices. Additionally, the bounds and error estimates would need to be redefined to account for the specific characteristics of non-symmetric or non-positive definite operators.

The techniques developed in this work for the infinite-dimensional Nyström approximation can potentially be applied to other randomized matrix approximation methods, such as the CUR decomposition, in the infinite-dimensional setting. The key lies in adapting the analysis and bounds to suit the specific characteristics and requirements of the CUR decomposition. This would involve considering the sampling and approximation strategies unique to the CUR decomposition, as well as the properties of the matrices involved in the decomposition. By extending the techniques and methodologies developed for the Nyström approximation to the CUR decomposition, it is possible to derive error bounds and guarantees for the CUR approximation in infinite dimensions.

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