Towards persistence-based reconstruction in euclidean spaces

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recent advances in higher dimensions have led to new methods to reconstruct large classes of compact subsets of Rd . However, the complexities of these methods scale up exponentially with d, making them impractical in medium or high dimensions, even on data sets of low intrinsic dimensionality. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Our algorithm combines two paradigms: greedy refinement, and topological persistence. Given a point cloud in Rd , we build a set of landmarks iteratively, while maintaining a nested abstract complexes, whose images in Rd lie close to the data, and whose persistent homology eventually coincides with the homology of the underlying shape. When the data points are densely sampled from a smo...

• Type: Conference
• Year: 2008
• Where: COMPGEOM
• Authors: Frédéric Chazal, Steve Oudot
• Keywords: COMPGEOM 2008, Discrete Geometry, Euclidean Spaces, Low Intrinsic Dimensionality, Witness Complex Filtrations