Non-asymptotic Error Bound for Optimal Prediction of Function-on-Function Regression by RKHS Approach

Domain Generalization by Functional Regression

“…This enables to estimate the regression functions in each domain by different kernels, and, to apply explicit finite-sample bounds from the field of functional regression, e.g. Mollenhauer et al (2022); Jin et al (2022);Tong et al (2022). The independence of the noise is for simplicity and can be removed at the price of slightly more involved proofs, see (Jin et al, 2022, Eq. (1)).…”

“…( 13). In this step, we follow stepby-step the Algorithm 1 in Section 4 of Tong et al (2022). This algorithm is essentially ridge regression, but it requires to estimate functions instead of scalar weights for the kernel sections in the solution granted by the representer theorem, see e.g.…”

“…In order to prove Lemma 9, we will discuss the following Corollary first, which applies the main arguments from Tong et al (2022) to our setting. For readers convenience, we provide an overview here:…”

“…More precisely, we propose a new algorithm for explicitly learning (from the input samples) an operator, which maps (kernel mean embeddings of) input marginal distributions P X (x) to approximations of the regression functions (Bayes predictors) f P := arg min f :X →Y X ×Y (f (x) − y) 2 dP (x, y) of the corresponding conditional distributions P (y|x). We, here, in a first step, focus on learning a linear operator with slope functions residing in an RKHS, which allows us to apply analytical arguments from Mollenhauer et al (2022); Jin et al (2022); Tong et al (2022) resulting in explicit finite sample bounds. However, our new concept opens new directions for domain generalization by linear and non-linear operator learning.…”