Convergence Analysis of function-on-function Polynomial regression model

TL;DR

This paper analyzes the convergence properties of a regularization-based algorithm for function-on-function polynomial regression in infinite-dimensional Hilbert spaces, providing convergence rates and optimality bounds.

Contribution

It introduces a general spectral regularization framework for analyzing convergence in infinite-dimensional settings without extra assumptions.

Findings

Derived convergence rates for estimation and prediction errors.

Established lower bounds demonstrating the optimality of the rates.

Provided a comprehensive analysis applicable to a broad class of regularization methods.

Abstract

In this article, we study the convergence behavior of the regularization-based algorithm for solving the polynomial regression model when both input data and responses are from infinite-dimensional Hilbert spaces. We derive convergence rates for estimation and prediction error by employing general (spectral) regularization under a general smoothness condition without imposing any additional conditions on the index function. We also establish lower bounds for any learning algorithm to explain the optimality of our convergence rates.