Colloque - Anthony Nouy : Stable Nonlinear Manifold Approximation Using Compositional Networks

Yvon Maday

Chaire Informatique et sciences numériques

Collège de France

Année 2025-2026

Colloque : Aspects mathématiques et appliqués des méthodes de réduction de complexité - Anthony Nouy : Stable Nonlinear Manifold Approximation Using Compositional Networks

Anthony Nouy

Professeur au département de mathématiques, Centrale Nantes – Nantes Université

Résumé

We consider the problem of approximating a subset M of a Hilbert space X by a low-dimensional manifold Mn. A large class of nonlinear methods can be described by a decoder D: IRn à X whose range is the nonlinear manifold Mn, and an encoder E : E à IRn 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 worst-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.

Joint work with A. Bensalah, J. Soffo, A. Somacal.