Dimensionality reduction for k-distance applied to persistent homology

Abstract

Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy (The persistent homology of distance functions under random projection. In Cheng, Devillers (eds), 30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08–11, p 328, ACM, 2014). We show that any linear transformation that preserves pairwise distances up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of $$(1-{\varepsilon })^{-1}$$ ( 1 - ε ) - 1 . Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. [J Comput Geom, 58:70–96, 2016] are preserved up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) factor. We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional submanifold, obtaining embeddings having the dimension bounds of Lotz (Proc R Soc A Math Phys Eng Sci, 475(2230):20190081, 2019) and Clarkson (Tighter bounds for random projections of manifolds. In Teillaud (ed) Proceedings of the 24th ACM Symposium on Computational Geom- etry, College Park, MD, USA, June 9–11, pp 39–48, ACM, 2008) respectively. Our results also work in the terminal dimensionality reduction setting, where the distance of any point in the original ambient space, to any point in P, needs to be approximately preserved.

Shape-Preserving Dimensionality Reduction : An Algorithm and Measures of Topological Equivalence

“…Moreover, Sheehy (2014) showed that a certain random projection preserving critical points also preserves the persistent homology. Along with this result, Arya et al (2020) shows that every linear projection map preserves the Čech complex arising from a specially weighted distance between a point and its point cloud.…”

“…Our approach is different from that of Arya et al (2020) whose goal is to find an "optimized" distance-preserving Čech complex working for all linear projections. On the other hand, we center on finding an "optimized" linear projection that preserves the Čech complex under the Euclidean distance.…”

Dimensionality Reduction for Persistent Homology with Gaussian Kernels

“…However, their techniques involve only the usual distance to a point set and therefore are highly sensitive to the presence of outliers and noise as mentioned earlier. The question of adapting the method of random projections in order to reduce the complexity of computing PH using the k-distance is therefore a natural one, and was addressed by Arya et al in [4] who showed that under random projections, the same bounds apply for the preservation of PH using the k-distance, as for pairwise distances.…”

“…preserving kernel distances between point distributions), it is not clear that it can preserve the PH, since this involves preserving intersections of multiple balls under a power distance (see e.g. [4,32]). A key issue under the GKPD is that the weights associated to the data points are not just a function of the points themselves, but of the pairwise kernel distances of all the points in the data set.…”

“…A similar problem arises in the case of computing the persistent homology under the k-distance. Addressing this was asked as an open problem in [35], and recently answered by Arya et al [4]. However, their solution crucially uses the linearity of the dimensionality-reducing map as well as certain properties of minimum enclosing balls under the k-distance, which they prove, and hence does not apply to the GKPD.…”