We consider the problem of approximating a subset $M$ of a Banach space by a low-dimensional manifold $M_n$. A large class of nonlinear methods can be described by a decoder $D$ mapping from $R^n$ to $X$ and whose
range is the nonlinear manifold $M_n$, and an encoder $E$ from $M$ to $R^n$ which extracts $n$ pieces of information $E(u)$ from an element $u$ in $M$. Here, we introduce a nonlinear method where $E$ is linear and $D$ is a stable decoder 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 a decoder which 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. Also, we discuss on the definition of optimal encoders and provide concrete strategies for their estimation.

This is joint work with Antoine Bensallah and Joel Soffo.

Reference: A. Bensalah, A. Nouy, and J. Soffo (2025). Nonlinear manifold approximation using compositional polynomial networks. arXiv e-prints arXiv:2502.05088, Feb. 2025.