# Approximate Nearest Neighbors in the Space of Persistence Diagrams

arXiv: https://arxiv.org/abs/1812.11257 · HTML: https://mlsanity.com/paper/1812.11257/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2018.

## Built on by (most cited first, 1 of 1)

- [A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering](https://mlsanity.com/paper/2105.12208/) (2021)

## Abstract (arXiv)

Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a set of other persistence diagrams is linear in the number of diagrams or do not offer performance guarantees. In this paper, we apply concepts from locality-sensitive hashing to support approximate nearest neighbor search in the space of persistence diagrams. Given a set Gamma of n (M,m)-bounded persistence diagrams, each with at most m points, we snap-round the points of each diagram to points on a cubical lattice and produce a key for each possible snap-rounding. Specifically, we fix a grid over each diagram at several resolutions and consider the snap-roundings of each diagram to the four nearest lattice points. Then, we propose a data structure with τ levels mathbbDτ that stores all snap-roundings of each persistence diagram in Gamma at each resolution. This data structure has size O(n5mτ) to account for varying lattice resolutions as well as snap-roundings and the deletion of points with low persistence. To search for a persistence diagram, we compute a key for a query diagram by snapping each point to a lattice and deleting points of low persistence. Furthermore, as the lattice parameter decreases, searching our data structure yields a six-approximation of the nearest diagram in Gamma in O((mlogn+m2)logτ) time and a constant factor approximation of the kth nearest diagram in O((mlogn+m2+k)logτ) time.