Persistent homology for low-complexity models

Abstract

We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated to the data set, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. This allows to literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.

Dimensionality reduction for k-distance applied to persistent homology

“…It is not very hard to see that for a given point set P, the random Johnson-Lindenstrauss mapping preserves the pointwise k-distance to P (Theorem 17). However, this is not enough to preserve intersections of balls at varying scales of the radius parameter, and thus does not suffice to preserve the persistent homology of Čech filtrations, as noted by Sheehy (2014) and Lotz (2019). We show how the squared radius of a set of weighted points can be expressed as a convex combination of pairwise squared distances.…”

“…The JL Lemma has also been used by Sheehy (2014) and Lotz (2019) to reduce the complexity of computing persistent homology. Both Sheehy and Lotz show that the persistent homology of a point cloud is approximately preserved under random projections (Sheehy 2014;Lotz 2019), up to a (1 ± ε) multiplicative factor, for any ε ∈ [0, 1].…”

“…The JL Lemma has also been used by Sheehy (2014) and Lotz (2019) to reduce the complexity of computing persistent homology. Both Sheehy and Lotz show that the persistent homology of a point cloud is approximately preserved under random projections (Sheehy 2014;Lotz 2019), up to a (1 ± ε) multiplicative factor, for any ε ∈ [0, 1]. Sheehy proves this for an n-point set, whereas Lotz's generalization applies to sets of bounded Gaussian width, and also implies dimensionality reductions for sets of bounded doubling dimension, in terms of the spread (ratio of the maximum to minimum interpoint distance).…”

“…The extensions provide bounds which do not depend on the number of points in the sample. The first one, analogous to Lotz (2019), shows that the persistent homology with respect to the k-distance, of point sets contained in regions having bounded Gaussian width, can be preserved via dimensionality reduction, using an embedding with dimension bounded by a function of the Gaussian width. Another result is that for points lying in a lowdimensional submanifold of a high-dimensional Euclidean space, the dimension of the embedding preserving the persistent homology with k-distance depends linearly on the dimension of the submanifold.…”

“…Run-time and Efficiency In many other applications of the Johnson-Lindenstrauss dimensionality reduction, multiplying by a dense gaussian matrix is a significant overhead, and can seriously affect any gains resulting from working in a lower dimensional space. However, as is pointed out in Lotz (2019), in the computation of persistent homology the dimensionality reduction step is carried out only once for the n data points at the beginning of the construction. Having said that, it should still be observed that most of the recent results on dimensionality reduction using sparse subgaussian matrices (Ailon and Chazelle 2009; Kane and Nelson 2014;Krahmer and Ward 2011) can also be used to compute the k-distance persistent homology, with little to no extra cost.…”