# Approximate Nearest Neighbors in the Space of Persistence Diagrams

@article{Fasy2018ApproximateNN, title={Approximate Nearest Neighbors in the Space of Persistence Diagrams}, author={Brittany Terese Fasy and Xiaozhou He and Zhihui Liu and Samuel Micka and David L. Millman and Binhai Zhu}, journal={ArXiv}, year={2018}, volume={abs/1812.11257} }

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…

## 7 Citations

Sketching Persistence Diagrams

- Computer Science, MathematicsSoCG
- 2021

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.

On Computing a Center Persistence Diagram

- Mathematics, Computer ScienceArXiv
- 2019

It is shown, by a non-trivial reduction from the Planar 3D-Matching problem, that this problem is NP-hard even when $m=3$ diagrams are given, which implies that the general center problem for persistence diagrams under the bottleneck distance, when $P_i$'s possibly have different sizes, is also NP- hard.

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

- Computer Science2022 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX)
- 2022

An open source software called PDoptFlow is developed based on the algorithm, exploiting parallelism by GPU and multicore, and it is shown that it can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.

Indexing Point Sets for Approximate Bottleneck Distance Queries

- Computer Science, MathematicsArXiv
- 2018

The main contribution of this work is a {\em trie}-based data structure that is space efficient and supports $8-approximate nearest bottleneck queries in $O(-\lg(d_B(D,Q)) n \lg^3 n)$ time, which is a simple time algorithm to support nearest bottleneck distance queries for any two point sets P and Q.

Computational Geometry Column 70: Processing Persistence Diagrams as Purely Geometric Objects

- 2020

In this column, we review the most recent results on processing persistence diagrams as purely geometric objects. (Yes, this is not a typo! While persistence diagrams originally come from…

Computational Geometry Column 70

- Computer ScienceSIGACT News
- 2020

In this column, the most recent results on processing persistence diagrams as purely geometric objects are reviewed, and the approximate nearest neighbor query under the bottleneck distance is reviewed.

A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering

- Computer Science, MedicineIEEE Transactions on Visualization and Computer Graphics
- 2022

A persistence diagram hashing framework that learns a binary code representation of persistence diagrams, which allows for fast computation of distances, and demonstrates that the method is significantly faster with the potential of less memory usage, while retaining comparable or better quality comparisons.

## References

SHOWING 1-10 OF 31 REFERENCES

Locality-Sensitive Hashing of Curves

- Computer Science, MathematicsSoCG
- 2017

The first locality-sensitive hashing schemes for these distance measures using the discrete Frechet distance or the dynamic time warping distance are devised, which provide a trade-off between approximation quality and computational performance.

Geometry Helps to Compare Persistence Diagrams

- Computer ScienceACM J. Exp. Algorithmics
- 2017

This work implements geometric variants of the Hopcroft-Karp algorithm for bottleneck matching and of the auction algorithm by Bertsekas for Wasserstein distance computation that lead to a substantial performance gain over their purely combinatorial counterparts.

Learning Simplicial Complexes from Persistence Diagrams

- Mathematics, Computer ScienceCCCG
- 2018

This paper presents an algorithm for reconstructing plane graphs K=(V,E) in R^2, i.e., a planar graph with vertices in general position and a straight-line embedding, from a quadratic number height filtrations and their respective persistence diagrams.

Approximate nearest neighbor algorithms for Frechet distance via product metrics

- Computer Science, MathematicsSCG '02
- 2002

Several data structures using space (quasi)-polynomial in n and d, and query time sublinear in n, have been discovered for approximate NNS under l1 and l2 [14, 12, 11] and l1 [10] norms.

Persistent Homology Transform for Modeling Shapes and Surfaces

- Mathematics
- 2013

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence…

Comparing Persistence Diagrams Through Complex Vectors

- Mathematics, Computer ScienceICIAP
- 2015

This article explores experimentally three transformations from diagrams to polynomials and three distances between the complex vectors of coefficients.

Geometry Helps in Bottleneck Matching and Related Problems

- Mathematics, Computer ScienceAlgorithmica
- 2001

An O(n5log n) -time algorithm for determining whether for some translated copy the resemblance gets below a given ρ is presented, thus improving the previous result of Alt, Mehlhorn, Wagener, and Welzl by a factor of almost n.

Approximate nearest neighbors: towards removing the curse of dimensionality

- Mathematics, Computer ScienceSTOC '98
- 1998

Two algorithms for the approximate nearest neighbor problem in high-dimensional spaces are presented, which require space that is only polynomial in n and d, while achieving query times that are sub-linear inn and polynometric in d.

Local persistent homology based distance between maps

- Mathematics, Computer ScienceSIGSPATIAL/GIS
- 2014

A topology-based distance metric between road networks embedded in the plane is defined, based on local persistent homology, and employs a local distance signature that enables identification and visualization of local differences between the road networks.

The Structure and Stability of Persistence Modules

- Mathematics, Computer ScienceSpringer Briefs in Mathematics
- 2016

This book is a comprehensive treatment of the theory of persistence modules over the real line. It presents a set of mathematical tools to analyse the structure and to establish the stability of such…