Estimation of the Mean Function of Functional Data via Deep Neural Networks

Abstract

In this work, we propose a deep neural network method to perform nonparametric regression for functional data. The proposed estimators are based on sparsely connected deep neural networks with ReLU activation function. By properly choosing network architecture, our estimator achieves the optimal nonparametric convergence rate in empirical norm. Under certain circumstances such as trigonometric polynomial kernel and a sufficiently large sampling frequency, the convergence rate is even faster than root-n rate. Through Monte Carlo simulation studies we examine the finite-sample performance of the proposed method. Finally, the proposed method is applied to analyze positron emission tomography images of patients with Alzheimer disease obtained from the Alzheimer Disease Neuroimaging Initiative database.

Calibrating multi-dimensional complex ODE from noisy data via deep neural networks

“…Owing to the superior performance of deep learning in modeling complicated data, deep neural networks have been actively used to reproduce dynamical systems ([Weinan, 2017]). Deep neural networks, such as residual network ([He et al, 2016]) and discrete normalizing flows ([Kobyzev et al, 2020]), can be considered as discrete dynamical systems.…”

CovNet: Covariance Networks for Functional Data on Multidimensional Domains

“…Neural networks have long been successfully used in nonparametric function estimation, and recently, they have been shown to successfully overcome the curse of dimensionality in nonparametric regression (Bauer and Kohler, 2019;Schmidt-Hieber, 2020). Also, they have been recently used for mean estimation of functional data over multidimensional domains (Wang, Cao and Shang, 2020). Motivated by the success of neural networks, we propose Covariance Networks (CovNet) as a framework for the estimation of the covariance of multidimensional random fields.…”