# Nonlinear Approximation via Compositions

## TL;DR

This paper explores how using compositions of functions in dictionaries, implemented via neural networks, can significantly improve nonlinear approximation rates, especially when increasing the number of composition layers from 1 to 3.

## Contribution

It introduces a novel approach of using compositional dictionaries with neural networks to enhance approximation rates and analyzes the impact of the number of layers on approximation efficiency.

## Key findings

- Increasing layers from 1 to 2 or 3 improves approximation rates.
- For functions on [0,1], L=2 achieves twice the rate of L=1.
- L=3 achieves near-optimal rates for Hölder continuous functions.

## Abstract

Given a function dictionary $\cal D$ and an approximation budget $N\in\mathbb{N}^+$, nonlinear approximation seeks the linear combination of the best $N$ terms $\{T_n\}_{1\le n\le N}\subseteq{\cal D}$ to approximate a given function $f$ with the minimum approximation error\[\varepsilon_{L,f}:=\min_{\{g_n\}\subseteq{\mathbb{R}},\{T_n\}\subseteq{\cal D}}\|f(x)-\sum_{n=1}^N g_n T_n(x)\|.\]Motivated by recent success of deep learning, we propose dictionaries with functions in a form of compositions, i.e.,\[T(x)=T^{(L)}\circ T^{(L-1)}\circ\cdots\circ T^{(1)}(x)\]for all $T\in\cal D$, and implement $T$ using ReLU feed-forward neural networks (FNNs) with $L$ hidden layers. We further quantify the improvement of the best $N$-term approximation rate in terms of $N$ when $L$ is increased from $1$ to $2$ or $3$ to show the power of compositions. In the case when $L>3$, our analysis shows that increasing $L$ cannot improve the approximation rate in terms of $N$. In particular, for any function $f$ on $[0,1]$, regardless of its smoothness and even the continuity, if $f$ can be approximated using a dictionary when $L=1$ with the best $N$-term approximation rate $\varepsilon_{L,f}={\cal O}(N^{-\eta})$, we show that dictionaries with $L=2$ can improve the best $N$-term approximation rate to $\varepsilon_{L,f}={\cal O}(N^{-2\eta})$. We also show that for H\"older continuous functions of order $\alpha$ on $[0,1]^d$, the application of a dictionary with $L=3$ in nonlinear approximation can achieve an essentially tight best $N$-term approximation rate $\varepsilon_{L,f}={\cal O}(N^{-2\alpha/d})$. Finally, we show that dictionaries consisting of wide FNNs with a few hidden layers are more attractive in terms of computational efficiency than dictionaries with narrow and very deep FNNs for approximating H\"older continuous functions if the number of computer cores is larger than $N$ in parallel computing.