Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Non-Parametric Estimation of Manifolds from Noisy Data   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 $\mathbb{R}^D$ from
a finite set of noisy samples. Assuming that the data was sampled uniformly
from a tubular neighborhood of $\mathcal{M}\in \mathcal{C}^k$, a compact
manifold without boundary, we present an algorithm that takes a point $r$ from
the tubular neighborhood and outputs $\hat p_n\in \mathbb{R}^D$, and
$\widehat{T_{\hat p_n}\mathcal{M}}$ an element in the Grassmanian $Gr(d, D)$.
We prove that as the number of samples $n\to\infty$ the point $\hat p_n$
converges to $p\in \mathcal{M}$ and $\widehat{T_{\hat p_n}\mathcal{M}}$
converges to $T_p\mathcal{M}$ (the tangent space at that point) with high
probability. Furthermore, we show that the estimation yields asymptotic rates
of convergence of $n^{-\frac{k}{2k + d}}$ for the point estimation and
$n^{-\frac{k-1}{2k + d}}$ for the estimation of the tangent space. These rates
are known to be optimal for the case of function estimation.

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The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: supr-con