The skew mean curvature flow is an evolution equation for
We consider the regression problem of estimating functions on $ \mathbb{R}^D $ but supported on a $ d $dimensional manifold $ \mathcal{M} ~~\subset \mathbb{R}^D $ with $ d \ll D $. Drawing ideas from multiresolution analysis and nonlinear approximation, we construct lowdimensional coordinates on $ \mathcal{M} $ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a datadriven wavelet thresholding scheme that automatically adapts to the unknown regularity of the function, allowing for efficient estimation of functions exhibiting nonuniform regularity at different locations and scales. We analyze the generalization error of our method by proving finite sample bounds in high probability on rich classes of priors. Our estimator attains optimal learning rates (up to logarithmic factors) as if the function was defined on a known Euclidean domain of dimension $ d $, instead of an unknown manifold embedded in $ \mathbb{R}^D $. The implemented algorithm has quasilinear complexity in the sample size, with constants linear in $ D $ and exponential in $ d $. Our work therefore establishes a new framework for regression on lowdimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.
 NSFPAR ID:
 10296149
 Date Published:
 Journal Name:
 Mathematics in Engineering
 Volume:
 4
 Issue:
 4
 ISSN:
 26403501
 Page Range / eLocation ID:
 1 to 25
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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