  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.
