Convergence Analysis of function-on-function Polynomial regression model
Naveen Gupta, Sivananthan Sampath
TL;DR
The paper develops a convergence theory for function-on-function polynomial regression under regularization, addressing the ill-posed nature of the problem when both inputs and outputs are infinite-dimensional. By leveraging a general spectral regularization framework and a generalized source condition, it obtains upper bounds for estimation and prediction errors that depend on the effective dimension and the smoothness imposed by an index function, and it proves optimality via matching lower bounds constructed through KL-divergence arguments. The results extend prior scalar-rooted findings to non-compact operators and all Hölder-type smoothness, offering rigorous guarantees for function-valued regression in FDA contexts. Overall, the work advances the understanding of non-linear, function-valued regression with non-Hilbert-Schmidt operators and informs practical regularization choices in high-dimensional functional data analysis.
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.