Nonparametric function-on-scalar regression using deep neural networks

Abstract

We focus on nonlinear Function-on-Scalar regression, where the predictors are scalar variables, and the responses are functional data. Most existing studies approximate the hidden nonlinear relationships using linear combinations of basis functions, such as splines. However, in classical nonparametric regression, it is known that these approaches lack adaptivity, particularly when the true function exhibits high spatial inhomogeneity or anisotropic smoothness. To capture the complex structure behind data adaptively, we propose a simple adaptive estimator based on a deep neural network model. The proposed estimator is straightforward to implement using existing deep learning libraries, making it accessible for practical applications. Moreover, we derive the convergence rates of the proposed estimator for the anisotropic Besov spaces, which consist of functions with varying smoothness across dimensions. Our theoretical analysis shows that the proposed estimator mitigates the curse of dimensionality when the true function has high anisotropic smoothness, as shown in the classical nonparametric regression. Numerical experiments demonstrate the superior adaptivity of the proposed estimator, outperforming existing methods across various challenging settings. Moreover, the proposed method is applied to analyze ground reaction force data in the field of sports medicine, demonstrating more efficient estimation compared to existing approaches.