Papers › Non-Parametric Estimation of Manifolds from Noisy Data
Non-Parametric Estimation of Manifolds from Noisy Data
Yariv Aizenbud, Barak Sober
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 ℝᴰ from a finite set of noisy samples. Assuming that the data was sampled uniformly from a tubular neighborhood of ℳ∈𝒞ᵏ, a compact manifold without boundary, we present an algorithm that takes a point r from the tubular neighborhood and outputs p̂ₙ∈ℝᴰ, and T_(p̂ₙ)ℳ an element in the Grassmanian Gr(d, D). We prove that as the number of samples n→∞ the point p̂ₙ converges to p∈ℳ and T_(p̂ₙ)ℳ converges to Tₚℳ (the tangent space at that point) with high probability. Furthermore, we show that the estimation yields asymptotic rates of convergence of n^(-k/(2k + d)) for the point estimation and n^(-(k-1)/(2k + d)) for the estimation of the tangent space. These rates are known to be optimal for the case of function estimation.
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