"Instability of Manifold Reach and a Stable Combinatorial Reach"Rawson, MichaelIn application and theory, reconstruction of an embedded manifold from samples hinges on the manifold's $curvature$ and $reach$. The $reach$ of a manifold is the smallest distance to the medial axis. Reach is a function of curvature but contains more information for reconstruction. We show instability, theoretically and practically, in Federer's manifold reach formula. To compensate for this defect, we introduce a $combinatorial\ reach$ for stratifications. We show the reach and the combinatorial reach are equivalent in the limit of sampling on a manifold. With high probability, the sampling rate necessary to detect reach via combinatorial reach is a function of reach class and multiplicity. With noisy samples, we show that there is a trade-off in estimation error of the $combinatorial\ reach$ based on the noise covariance matrix and the curvature of the manifold. Noisy samples corrupt estimates of the curvature so we are forced to estimate both the reach and noise level by cross validation. We compare these noise level estimates with those obtained by piece-wise linear methods. We find the combinatorial reach noise estimates far more suitable for high dimensional, high curvature, or sparse data problems. We showcase these results on samples of conic sections, since conic sections are second order approximations of smooth manifolds locally. |
| http://univie.ac.at/projektservice-mathematik/e/talks/Rawson_2021-06_Online_ICCHA_2021_Rawson (1).pdf |
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