Topologically penalized regression on manifolds

Olympio Hacquard (1, 2) , Krishnakumar Balasubramanian (3) , Gilles Blanchard (1, 2) , Wolfgang Polonik (3) , Clément Levrard (4)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 DATASHAPE - Understanding the Shape of Data
3 UC Davis - University of California [Davis]
4 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is "topologically smooth".

Mots clés: Manifold regression; Statistical learning; Graph Laplacian; Topological persistence; Penalized models

Domaines: Statistiques [math.ST]; Apprentissage [cs.LG]

Olympio Hacquard, Krishnakumar Balasubramanian, Gilles Blanchard, Wolfgang Polonik, Clément Levrard. Topologically penalized regression on manifolds. 2021. ⟨hal-03402076v1⟩