Clément Berenfeld (UP)
Given i.i.d. random variables drawn from a mixture of immersed low-dimensional submanifolds of different dimensions, we study the minimax estimation of the family of submanifolds in Hausdorff distance. 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, the dimensions of these layers, assign each data point to a layer accurately, and estimate tangent spaces optimally.
These results hold regardless of any reach assumption on the submanifolds or on their (auto) intersection configurations.
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