Do Neural Networks Really Beat the Curse of Dimensionality? A Bit-Complexity View
Traditional approximation theory measures convergence rates in terms of the number of parameters or degrees of freedom. However, practical computation operates under finite precision: parameters must be encoded using a finite number of bits. Therefore, approximation efficiency should be evaluated in terms of computational bit complexity, which is intrinsically connected to the metric entropy of the underlying function class. In this work, we develop a unified approximation framework based on binary encoding and metric entropy. We analyze classical methods (including polynomial approximation, sparse grids, and finite elements) as well as shallow and deep neural networks, and compare their ...
| Authors | Tong Mao, Jinchao Xu |
|---|---|
| Primary category | cs.LG |
| Published | 2 August 2026 |
| Last updated at source | 2 August 2026 |
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