Minimax optimality of convolutional neural networks for infinite dimensional input-output problems and separation from kernel methods

Recent deep learning applications, exemplified by text-to-image tasks, often involve high-dimensional inputs and outputs. While several studies have investigated the function estimation capabilities of deep learning, research on dilated convolutional neural networks (CNNs) has mainly focused on cases where input dimensions are infinite but output dimensions are one-dimensional, similar to many other studies. However, many practical deep learning tasks involve highdimensional (or even infinite dimensional) inputs and outputs. In this paper, we investigate the optimality of dilated CNNs for estimating a map between infinitedimensional input and output spaces by analyzing their approximation and estimation abilities. For that purpose, we first show that approximation and estimation errors depend only on the smoothness and decay rate with respect to the infinity norm of the output, and their estimation accuracy actually achieve the minimax optimal rate of convergence. Second, we demonstrate that the dilated CNNs outperform any linear estimators including kernel ridge regression and k-NN estimators in a minimax error sense, highlighting the usefulness of feature learning realized by deep neural networks. Our theoretical analysis provide a theoretical basis for understanding the success of deep learning in recent high-dimensional input-output tasks. 1. We consider a nonlinear operator as a true function with infinite-dimensional inputs and outputs. In the aforementioned setup, we demonstrate that dilated CNNs achieve approximation and estimation errors that depend only on the smoothness and decay rate of the output. Furthermore,