Repository: Freie Universit├Ąt Berlin, Math Department

Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm

Li, Z. and Meunier, D. and Mollenhauer, M. and Gretton, A. (2023) Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm. arXiv . (Submitted)

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Official URL: https://doi.org/10.48550/arXiv.2312.07186

Abstract

We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between L2 and the hypothesis space, which we consider as a vector-valued reproducing kernel Hilbert space. These rates allow to treat the misspecified case in which the true regression function is not contained in the hypothesis space. We combine standard assumptions on the capacity of the hypothesis space with a novel tensor product construction of vector-valued interpolation spaces in order to characterize the smoothness of the regression function. Our upper bound not only attains the same rate as real-valued kernel ridge regression, but also removes the assumption that the target regression function is bounded. For the lower bound, we reduce the problem to the scalar setting using a projection argument. We show that these rates are optimal in most cases and independent of the dimension of the output space. We illustrate our results for the special case of vector-valued Sobolev spaces

Item Type:Article
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
Divisions:Department of Mathematics and Computer Science > Institute of Mathematics > Deterministic and Stochastic PDEs Group
ID Code:3092
Deposited By: Ulrike Eickers
Deposited On:19 Feb 2024 13:39
Last Modified:19 Feb 2024 13:39

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