Estimation of the Mean Function of Functional Data via Deep Neural Networks
This work addresses the problem of efficiently estimating the mean function of functional data, which is relevant for researchers working with high-dimensional or continuous data streams, such as medical imaging.
This paper proposes a deep neural network method for nonparametric regression of functional data. The method achieves the optimal nonparametric convergence rate in empirical norm, and under specific conditions, it can even surpass the root-n rate.
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.