Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homotopy Reconstruction via the Cech Complex and the Vietoris-Rips
  Complex   We derive conditions under which the reconstruction of a target space is
topologically correct via the \v{C}ech complex or the Vietoris-Rips complex
obtained from possibly noisy point cloud data. We provide two novel theoretical
results. First, we describe sufficient conditions under which any non-empty
intersection of finitely many Euclidean balls intersected with a positive reach
set is contractible, so that the Nerve theorem applies for the restricted
\v{C}ech complex. Second, we demonstrate the homotopy equivalence of a positive
$\mu$-reach set and its offsets. Applying these results to the restricted
\v{C}ech complex and using the interleaving relations with the \v{C}ech complex
(or the Vietoris-Rips complex), we formulate conditions guaranteeing that the
target space is homotopy equivalent to the \v{C}ech complex (or the
Vietoris-Rips complex), in terms of the $\mu$-reach. Our results sharpen
existing results.

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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci