# 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… Expand

#### 7 Citations

Sketching Persistence Diagrams

- Computer Science, Mathematics
- SoCG
- 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. Expand

On Computing a Center Persistence Diagram

- Mathematics, Computer Science
- ArXiv
- 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. Expand

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

- Computer Science
- ArXiv
- 2021

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. Expand

Indexing Point Sets for Approximate Bottleneck Distance Queries

- Computer Science, Mathematics
- ArXiv
- 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. Expand

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… Expand

Computational Geometry Column 70

- Computer Science
- SIGACT 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. Expand

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

- Medicine, Computer Science
- IEEE transactions on visualization and computer graphics
- 2021

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. Expand

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