Approximation of Nonlinear Functionals Using Deep ReLU Networks
This work addresses a theoretical gap in functional neural networks for researchers in approximation theory and machine learning, though it appears incremental as it extends existing methods to ReLU activation.
The paper tackles the problem of approximating nonlinear continuous functionals on L^p spaces using deep ReLU networks, which lacked theoretical analysis for ReLU activation, by constructing a continuous piecewise linear interpolation and establishing approximation rates under mild regularity conditions.
In recent years, functional neural networks have been proposed and studied in order to approximate nonlinear continuous functionals defined on $L^p([-1, 1]^s)$ for integers $s\ge1$ and $1\le p<\infty$. However, their theoretical properties are largely unknown beyond universality of approximation or the existing analysis does not apply to the rectified linear unit (ReLU) activation function. To fill in this void, we investigate here the approximation power of functional deep neural networks associated with the ReLU activation function by constructing a continuous piecewise linear interpolation under a simple triangulation. In addition, we establish rates of approximation of the proposed functional deep ReLU networks under mild regularity conditions. Finally, our study may also shed some light on the understanding of functional data learning algorithms.