Neural Approximation by Function Composition: Rigidity and Doubly Exponential Convergence
Deep neural networks approximate functions by composing affine maps with nonlinear activations, but how composition itself creates approximation power is not yet fully understood. We investigate a fundamental mechanism: geometrically weighted sums of iterates of a single scalar generator function. This mechanism underpins the classical tent-map construction of the function \(x - x^2\) and related recursive representations used by Yarotsky, W. E, et al., to analyze the approximation powers of deep neural networks. First, we establish a rigidity theorem: for continuous piecewise linear generators with a finite number of segments, any \(C^3\) function that can be represented in this way is a...
| Authors | Wentao Huang, Haizhang Zhang |
|---|---|
| Primary category | cs.LG |
| Published | 22 September 2026 |
| Last updated at source | 22 September 2026 |
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