On Explicit Super-Expressive Approximation for Neural Networks

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-maxD,4, depth-5 network with explicit parameter-error trade-offs. For H"older-smooth functions in Cr,γAleft([0,1]Dright), our fixed network of width max2D, D+5N+1 and depth r + 9 achieves the parameter magnitude P bounded by log2 P=Obigl(varepsilon-2D/(r+γ)log(1/varepsilon)bigr). This is the dual result compared to those in the parameter-bounded and architecture-unbounded paradigm.