Topological information retrieval with dilation-invariant bottleneck comparative measures
File(s)
Accepted version
Author(s)
Cao, Yueqi
Vlontzos, Athanasios
Schmidtke, Luca
Kainz, Bernhard
Monod, Anthea
Type
Journal Article
Abstract
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.
Date Issued
2021-04-04
Date Acceptance
2023-05-01
Citation
2021
ISSN
2049-8772
Publisher
Oxford University Press
Journal / Book Title
Information and Inference: a Journal of the IMA
Copyright Statement
Copyright reserved
Identifier
http://arxiv.org/abs/2104.01672v3
Subjects
68P15, 68P20, 55N31
cs.LG
math.AT
stat.ML
stat.ML
Notes
29 pages, 10 figures, 4 tables
Publication Status
Accepted