# Comparing persistence diagrams through complex vectors

arXiv: https://arxiv.org/abs/1505.01335 · HTML: https://mlsanity.com/paper/1505.01335/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2015.

## Built on by (most cited first, 9 of 9)

- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Learning Hyperbolic Representations of Topological Features](https://mlsanity.com/paper/2103.09273/) (2021)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [Smart Vectorizations for Single and Multiparameter Persistence](https://mlsanity.com/paper/2104.04787/) (2021)
- [Fixation Sequences as Time Series: A Topological Approach to Dyslexia Detection](https://mlsanity.com/paper/2604.21698/) (2026)
- [Convolutional Persistence Transforms](https://mlsanity.com/paper/2208.02107/) (2022)
- [Persistence Spheres: Bi-continuous Representations of Persistence Diagrams](https://mlsanity.com/paper/2509.16999/) (2025)
- [Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams](https://mlsanity.com/paper/2607.27126/) (2026)

## Abstract (arXiv)

The natural pseudo-distance of spaces endowed with filtering functions is precious for shape classification and retrieval; its optimal estimate coming from persistence diagrams is the bottleneck distance, which unfortunately suffers from combinatorial explosion. A possible algebraic representation of persistence diagrams is offered by complex polynomials; since far polynomials represent far persistence diagrams, a fast comparison of the coefficient vectors can reduce the size of the database to be classified by the bottleneck distance. This article explores experimentally three transformations from diagrams to polynomials and three distances between the complex vectors of coefficients.