Non-Parametric Estimation of Manifolds from Noisy Data

Author name not available (Why is that?)

Publication date: 10 May 2021

Abstract: A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. In this work, we consider the problem of estimating a

d

dimensional sub-manifold of

m a t h b b R^{D}

from a finite set of noisy samples. Assuming that the data was sampled uniformly from a tubular neighborhood of

m a t h c a l M i n m a t h c a l C^{k}

, a compact manifold without boundary, we present an algorithm that takes a point

r

from the tubular neighborhood and outputs

h a t p_{n} i n m a t h b b R^{D}

, and

w i d e h a t T_{h a t p_{n}} m a t h c a l M

an element in the Grassmanian

G r (d, D)

. We prove that as the number of samples

n o i n f t y

the point

h a t p_{n}

converges to

p i n m a t h c a l M

and

w i d e h a t T_{h a t p_{n}} m a t h c a l M

converges to

T_{p} m a t h c a l M

(the tangent space at that point) with high probability. Furthermore, we show that the estimation yields asymptotic rates of convergence of

n^{- f r a c k 2 k + d}

for the point estimation and

n^{- f r a c k - 1 2 k + d}

for the estimation of the tangent space. These rates are known to be optimal for the case of function estimation.

Has companion code repository: https://github.com/aizeny/manapprox

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