On Explicit Super-Expressive Approximation for Neural Networks

arXiv:2607.06781v1 Announce Type: new Abstract: In this work, we investigate the fixed-architecture neural network approximation with explicit parameter bounds and elementary activations. While prior work demonstrated super-expressive approximation using fixed-size networks, they lack quantitative and non-asymptotic characterizations of parameter magnitude with respect to the approximation error. We resolve this issue by introducing the Chinese Remainder Theorem as a constructive encoding mechanism. For Lipschitz continuous functions on [0,1]^D, we construct a width-max{D,4}, depth-5 network with explicit parameter-error trade-offs. For Holder-smooth functions in C^{r,gamma}_Aleft([0,1]^Dright), our fixed network of width max{2D, D+5N+1} and depth r + 9 achieves the parameter magnitude P bounded by log_2 P=Oigl(arepsilon^{-2D/(r+gamma)}log(1/arepsilon)igr). This is the dual result compared to those in the parameter-bounded and architecture-unbounded paradigm.

Source: arXiv cs.LG | 2026-07-09