Nonlinear manifold approximation using compositional polynomial networks

TL;DR

This paper introduces a nonlinear manifold approximation method using compositional polynomial networks, combining a linear encoder with a tree-structured polynomial decoder for efficient, stable, and accurate low-dimensional manifold representation.

Contribution

The paper presents a novel nonlinear approximation framework with a polynomial decoder composed via tree-structured composition, including error analysis and adaptive subspace construction.

Findings

Provides rigorous error and stability analysis.

Demonstrates effective numerical experiments.

Ensures controlled approximation and stability.

Abstract

We consider the problem of approximating a subset $M$ of a Hilbert space $X$ by a low-dimensional manifold $M_n$, using samples from $M$. We propose a nonlinear approximation method where $M_n $ is defined as the range of a smooth nonlinear decoder $D$ defined on $\mathbb{R}^n$ with values in a possibly high-dimensional linear space $X_N$, and a linear encoder $E$ which associates to an element from $ M$ its coefficients $E(u)$ on a basis of a $n$-dimensional subspace $X_n \subset X_N$, where $X_N$ is an optimal or near to optimal linear space, depending on the selected error measure The linearity of the encoder allows to easily obtain the parameters $E(u)$ associated with a given element $u$ in $M$. The proposed decoder is a polynomial map from $\mathbb{R}^n$ to $X_N$ which is obtained by a tree-structured composition of polynomial maps, estimated sequentially from samples in $M$. Rigorous error and stability analyses are provided, as well as an adaptive strategy for constructing the subspace $X_n$, and a decoder that guarantees an approximation of the set $M$ with controlled mean-squared or wort-case errors, and a controlled stability (Lipschitz continuity) of the encoder and decoder pair. We demonstrate the performance of our method through numerical experiments.