# Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures

arXiv: https://arxiv.org/abs/2104.01672 · HTML: https://mlsanity.com/paper/2104.01672/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 69 references that are in the map.
- First posted on arXiv 4 Apr 2021.

## Builds on (most cited first, 23 of 23)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [MedMNIST Classification Decathlon: A Lightweight AutoML Benchmark for Medical Image Analysis](https://mlsanity.com/paper/2010.14925/) (2020)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Connectivity-Optimized Representation Learning via Persistent Homology](https://mlsanity.com/paper/1906.09003/) (2019)
- [Geometry Helps to Compare Persistence Diagrams](https://mlsanity.com/paper/1606.03357/) (2016)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [Subsampling Methods for Persistent Homology](https://mlsanity.com/paper/1406.1901/) (2014)
- [Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport](https://mlsanity.com/paper/1901.03048/) (2019)
- [Adapting BERT for Word Sense Disambiguation with Gloss Selection Objective and Example Sentences](https://mlsanity.com/paper/2009.11795/) (2020)
- [A persistent homology approach to heart rate variability analysis with an application to sleep-wake classification](https://mlsanity.com/paper/1908.06856/) (2019)
- [Evasion Paths in Mobile Sensor Networks](https://mlsanity.com/paper/1308.3536/) (2013)
- [Cross-Lingual Document Retrieval with Smooth Learning](https://mlsanity.com/paper/2011.00701/) (2020)
- [Joint Geometric and Topological Analysis of Hierarchical Datasets](https://mlsanity.com/paper/2104.01395/) (2021)

## Built on by (most cited first, 2 of 2)

- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [A Normalized Bottleneck Distance on Persistence Diagrams and Homology Preservation under Dimension Reduction](https://mlsanity.com/paper/2306.06727/) (2023)

## Abstract (arXiv)

Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in a hierarchy-preserving manner using a variety of metrics. Persistent homology is a tool commonly used in topological data analysis that is able to rigorously characterize a database in terms of both its hierarchy and connectivity structure. Computing persistent homology on a variety of embedded datasets reveals that some commonly used embeddings fail to preserve the connectivity. We show that those embeddings which successfully retain the database topology coincide in persistent homology by introducing two dilation-invariant comparative measures to capture this effect: in particular, they address the issue of metric distortion on manifolds. We provide an algorithm for their computation that exhibits greatly reduced time complexity over existing methods. We use these measures to perform the first instance of topology-based information retrieval and demonstrate its increased performance over the standard bottleneck distance for persistent homology. We showcase our approach on databases of different data varieties including text, videos, and medical images.