Log In Sign Up

Multiscale regression on unknown manifolds

by   Wenjing Liao, et al.

We consider the regression problem of estimating functions on ℝ^D but supported on a d-dimensional manifold ℳ⊂ℝ^D with d ≪ D. Drawing ideas from multi-resolution analysis and nonlinear approximation, we construct low-dimensional coordinates on ℳ at multiple scales, and perform multiscale regression by local polynomial fitting. We propose a data-driven 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 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 low-dimensional sets embedded in high dimensions, with fast implementation and strong theoretical guarantees.


page 1

page 2

page 3

page 4


Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data

We consider the problem of efficiently approximating and encoding high-d...

Rates of Uniform Consistency for k-NN Regression

We derive high-probability finite-sample uniform rates of consistency fo...

Approximation of Functions on Manifolds in High Dimension from Noisy Scattered Data

In this paper, we consider the fundamental problem of approximation of f...

Koopman Methods for Estimation of Animal Motions over Unknown, Regularly Embedded Submanifolds

This paper introduces a data-dependent approximation of the forward kine...

Max-Affine Regression: Provable, Tractable, and Near-Optimal Statistical Estimation

Max-affine regression refers to a model where the unknown regression fun...

Wavelet eigenvalue regression in high dimensions

In this paper, we construct the wavelet eigenvalue regression methodolog...