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 pair of 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 smooth m-submanifold X of Rd, our method retrieves the homology of X in time at most c(m)n5, 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 filtrations in Euclidean spaces. More precisely, we show how previous results on unions of balls can be transposed to Čech filtrations, and from there to Rips and witness complex filtrations. Finally, investigating further on witness complexes, we quantify a conjecture of Carlsson and de Silva, which states that witness complex filtrations should have cleaner persistence barcodes than Čech or Rips filtrations, at least on smooth submanifolds of Euclidean spaces.