Group Seminar: Inverse Problems and Mathematical Imaging

Shuai Lu, Fudan University, Shanghai, China. Title: Function and derivative approximation by shallow neural networks

Thursday, August 21, 2024, 14:00
Zoom Meeting & University of Vienna, HS13, OMP1

Function and derivative approximation by shallow neural networks

Abstract: We investigate a Tikhonov regularization scheme specifically tailored for shallow neural networks within
the context of solving a classic inverse problem: approximating an unknown function and its derivatives
within a unit cubic domain based on noisy measurements. The proposed Tikhonov regularization scheme
incorporates a penalty term that takes three distinct yet intricately related network (semi)norms: the
extended Barron norm, the variation norm, and the Radon-BV seminorm. These choices of the penalty
term are contingent upon the specific architecture of the neural network being utilized. We establish the
connection between various network norms and particularly trace the dependence of the dimensionality
index, aiming to deepen our understanding of how these norms interplay with each other. We revisit the
universality of function approximation through various norms, establish rigorous error-bound analysis for
the Tikhonov regularization scheme, and explicitly elucidate the dependency of the dimensionality index,
providing a clearer understanding of how the dimensionality affects the approximation performance and
how one designs a neural network with diverse approximating tasks. It is a joint work with Yuanyuan Li
(Fudan University).

Zoom
Link: https://univienna.zoom.us/j/62860470950?pwd=OTRrk42rO0ZV88KPVtXPtbbOkzyUez.1