Geometric Inference for Probability Measures
- 752 Downloads
Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers or with background noise. In this paper, we show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we introduce a notion of distance function to a probability distribution in ℝ d . These functions share many properties with classical distance functions, which make them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, we show that it is possible to reconstruct offsets of sampled shapes with topological guarantees even in the presence of outliers. Moreover, in settings where empirical measures are considered, these functions can be easily evaluated, making them of particular practical interest.
KeywordsGeometric inference Computational topology Optimal transportation Nearest neighbor Surface reconstruction
Mathematics Subject Classification (2000)62G05 62-07 28A33
Unable to display preview. Download preview PDF.
- 5.F. Chazal, S.Y. Oudot, Towards persistence-based reconstruction in Euclidean spaces, in Proc. 24th ACM Sympos. Comput. Geom. (2008), pp. 232–241. Google Scholar
- 12.V. de Silva, G. Carlsson, Topological estimation using witness complexes, in Symposium on Point-Based Graphics, ETH, Zürich, Switzerland (2004). Google Scholar
- 17.K. Grove, Critical point theory for distance functions, in Proc. of Symposia in Pure Mathematics, vol. 54 (1993). Google Scholar
- 18.A. Lieutier, Any open bounded subset of ℝn has the same homotopy type as its medial axis, Comput. Aided Geom. Des. 36(11), 1029–1046 (2004). Google Scholar
- 20.P. Niyogi, S. Smale, S. Weinberger, A topological view of unsupervised learning from noisy data. Preprint (2008). Google Scholar
- 23.A. Petrunin, Semiconcave functions in Alexandrov’s geometry, in Surveys in differential geometry, vol. XI (International Press, Somerville, 2007), pp. 137–201. Google Scholar