Functional Equivalence and Geometric Diversity in Neural Network Approximations: An Empirical Characterization

תקציר מקורי באנגליתarXiv:2607.18930v1 Announce Type: new Abstract: The Universal Approximation Theorem states that a neural network with a single hidden layer is sufficient to approximate any continuous univariate function on a compact domain to arbitrary error. However, the uniqueness of such neural network representations is not guaranteed, raising questions about practical identifiability. In this work, we address this concern by analyzing functional equivalence and geometric diversity of neural network approximations to a few elementary mathematical functions. The analysis includes an extensive study of single-layer neural networks and multilayer perceptrons under noisy and noise-free conditions. Beyond just network capacity, we study the geometric properties through the lens of sloppiness, characterized