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 Double strok R sign^d. 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 Double strok R sign ^d, we build a set of landmarks iteratively, while maintaining a nested pair of abstract complexes, whose images in Double strok R sign ^d 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 smooth m-submanifold X of Double strok R sign ^d, our method retrieves the homology of X in time at most c(m)n ⁵, where n is the size of the input and c(m) is a constant depending solely on m. To prove the correctness of our algorithm, we investigate on Čech, Rips, and witness complex nitrations in Euclidean spaces. More precisely, we show how previous results on unions of balls can be transposed to Čech fitrations, and from there to Rips and witness complex fitrations. Finally, investigating further on witness complexes, we quantify a conjecture of Carlsson and de Silva, which states that witness complex fitrations should have cleaner persistence barcodes than Čech or Rips nitrations, at least on smooth sub-manifolds of Euclidean spaces.