A theory of stratification learning

TL;DR

This paper introduces a new adaptive algorithm for estimating the structure of stratified manifold mixtures from data, accurately identifying layers, dimensions, and tangent spaces without ambient assumptions.

Contribution

It presents a constructive, minimax-optimal method for stratification learning that adaptively detects layers and estimates their properties in complex manifold mixtures.

Findings

Algorithm achieves optimal dimension-specific rates

Successfully identifies number and dimensions of layers

Estimates tangent spaces accurately

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.