Nonlinear Manifold Approximation Using Compositional Polynomial Networks

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.