Understanding Smoothness of Vector Gaussian Processes on Product Spaces

Published: 13 May 2024, Last Modified: 13 May 2024Accepted by TMLREveryoneRevisionsBibTeX
Abstract: Vector Gaussian processes are becoming increasingly important in machine learning and statistics, with applications to many branches of applied sciences. Recent efforts have al- lowed to understand smoothness in scalar Gaussian processes defined over manifolds as well as over product spaces involving manifolds. Under assumptions of Gaussianity and mean-square continuity, the smoothness of a zero- mean scalar process is in one-to-one correspondence with the smoothness of the covariance kernel. Unfortunately, such a result is not available for vector-valued random fields, as the way each component in the covariance kernel contributes to the smoothness of the vector field is unclear. This paper challenges the problem of quantifying smoothness of matrix-valued continuous kernels that are associated with mean-square continuous vector Gaussian processes defined over non-Euclidean product manifolds. After noting that a constructive RKHS approach is unsuitable for this specific task, we proceed through the analysis of spectral properties. Specifically, we find a spectral representation to quantify smoothness through Sobolev spaces that are adapted to certain measure spaces of product measures obtained through the ten- sor product of Haar measures with multivariate Gaussian measures. Our results allow to measure smoothness in a simple way, and open for the study of foundational properties of certain machine learning techniques over product spaces.
Submission Length: Long submission (more than 12 pages of main content)
Previous TMLR Submission Url: https://openreview.net/forum?id=sNM4jUDoly&referrer=%5BAuthor%20Console%5D(%2Fgroup%3Fid%3DTMLR%2FAuthors%23your-submissions)
Changes Since Last Submission: Dear Editors, we have the pleasure to submit a revised version that addresses all the queries by the four reviewers. A reply to Reviewers has been included as Supplementary Material. We wish to note that we are using the template from TMLR. There was an initial note from the action Editor regarding the margins. We cannot see any difference with respect to the official template. Should the Editor have concerns regarding this aspect, please do not hesitate to contact me. Thank you so much, The authors
Supplementary Material: pdf
Assigned Action Editor: ~Peter_Richtarik1
Submission Number: 2007