Stable Nonlinear Manifold Approximation Using Compositional Networks

Abstract

We consider the problem of approximating a subset M of a Hilbert space X by a low-dimensional manifold M_n . A large class of nonlinear methods can be described by a decoder D : IRⁿ → X whose range is the nonlinear manifold M_n , and an encoder E : E → 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 obtained via a tree-structured composition of polynomial maps, estimated sequentially from samples in M . Rigorous error and stability analyses are provided, along with an adaptive strategy for constructing a decoder that guarantees an approximation of the set M with controlled mean-squared or worst-case errors, and controlled stability (Lipschitz continuity) of the encoder-decoder pair.

We also discuss the definition of optimal encoders and provide concrete strategies for their estimation.

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

Anthony Nouy

Anthony Nouy is a full professor in the Department of Mathematics at Centrale Nantes/Nantes Université. He received his PhD in 2003 from the École normale supérieure de Cachan and his habilitation in 2008. His research spans many areas, ranging from numerical analysis and approximation theory to statistics and numerical probability. He has made notable contributions to high-dimensional approximation and learning, particularly in model order reduction, tensor methods, and uncertainty quantification. He has served on the editorial boards of leading journals. He is Director of the French research network on UQ and of the activity group on Signal, Image, Geometry, Modeling, and Approximation of the French Applied Mathematics Society, fostering community interaction.