Compositional Approximation Can Strictly Outperform Superpositional Approximation

2026-06-07

AI Summary

This work investigates the comparative capabilities of compositional and additive approximation approaches under parameter efficiency constraints. By constructing function classes with specific structural properties and leveraging tools from function approximation theory, parameterized complexity analysis, and neural network encoding limitations, the study provides the first explicit example of functions for which compositional methods provably outperform additive ones. The analysis demonstrates that there exist function classes where compositional approximations achieve a strictly faster error decay rate than any additive method subject to bit-encoding constraints, and that the gap in approximation error between the two paradigms can be made arbitrarily large. These results formally establish the theoretical superiority of compositional approaches in certain regimes.

Abstract

Many classically studied function classes are known to be approximated optimally by superpositional methods, i.e. with approximants constructed as the linear combination of elements in some dictionary. Here optimality means that the uniform approximation error viewed as a function of the number of parameters used has polynomial decay of the highest order achievable by any parametrized method whose parameters can be encoded as a bit string of length proportional, up to logarithmic factors, to the number of parameters. While compositional methods like neural networks are structurally different, their approximation rates can be made comparable by imposing constraints that ensure such a proportional bit string encoding. In this work we study function classes exhibiting structural properties that limit superpositional approximation rates to be strictly lower than compositional approximation rates. In particular, we construct explicit examples for which there is an arbitrarily large gap.

Problem

• compositional approximation
• superpositional approximation
• approximation rates
• function classes
• structural properties

Innovation

• compositional approximation
• superpositional approximation
• approximation rates
• neural networks
• function classes