Sketching Persistence Diagrams

TL;DR

This paper introduces an efficient algorithm for sketching persistence diagrams, producing a sequence of approximations converging to the original diagram with guarantees on approximation quality and space complexity.

Contribution

The authors develop a novel sketching method for persistence diagrams that enables fast approximation of bottleneck distances and minimal space representation.

Findings

Sequence of diagrams converges in bottleneck distance

Space complexity of sketches is linear in the number of points

Allows linear-time approximation of Hausdorff and bottleneck distances

Abstract

Given a persistence diagram with $n$ points, we give an algorithm that produces a sequence of $n$ persistence diagrams converging in bottleneck distance to the input diagram, the $i$th of which has $i$ distinct (weighted) points and is a $2$-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the $i$th and the $(i+1)$st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in $O(n)$ space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams -- a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.