A theory of stratification learning

2024/05/30 by Aamari, Eddie, Berenfeld, Clément · 1 citation
#54E20 #57R42 #62G05 #Differential Geometry (math.DG) #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2405.20066

Abstract

Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we study the minimax estimation of the underlying stratified structure. We provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon.

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