# Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images

arXiv: https://arxiv.org/abs/2401.04332 · HTML: https://mlsanity.com/paper/2401.04332/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 49 references that are in the map.
- First posted on arXiv 9 Jan 2024.

## Builds on (most cited first, 6 of 6)

- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [An Introduction to Multiparameter Persistence](https://mlsanity.com/paper/2203.14289/) (2022)
- [Topologically penalized regression on manifolds](https://mlsanity.com/paper/2110.13749/) (2021)

## Built on by (most cited first, 1 of 1)

- [G-LoG Bi-filtration for Medical Image Classification](https://mlsanity.com/paper/2602.18329/) (2026)

## Abstract (arXiv)

Two important tasks in the field of Topological Data Analysis are building practical multifiltrations on objects and using TDA to detect the geometry. Motivated by the tasks, we build multiparameter filtrations by operators on images named multi-GENEO, multi-DGENEO and mix-GENEO, and we prove the stability of both the interleaving distance and multiparameter persistence landscape of multi-GENEO with respect to the pseudometric on bounded functions. We also give the estimations of upper bound for multi-DGENEO and mix-GENEO. In practical applications, we regard image as a discrete function space, and then we build multifiltrations on the discrete function space. Finally, we construct comparable experiment on MNIST dataset to demonstrate our bifiltrations are superior to 1-parameter filtrations including lower-star filtration and upper-star filtration. For instance, 6 and 9 can be distinguished by our bifiltrations, while they cannot be distinguished by 1-parameter filtrations. The experiment results demonstrate our bifiltrations have ability to detect geometric and topological differences of digital images.