# Filtration learning in exact multi-parameter persistent homology and classification of time-series data

arXiv: https://arxiv.org/abs/2406.19587 · HTML: https://mlsanity.com/paper/2406.19587/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 47 references that are in the map.
- First posted on arXiv 28 Jun 2024.

## Builds on (most cited first, 15 of 15)

- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [Understanding Notions of Stationarity in Non-Smooth Optimization](https://mlsanity.com/paper/2006.14901/) (2020)
- [Topological Time Series Analysis](https://mlsanity.com/paper/1812.05143/) (2018)
- [Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds](https://mlsanity.com/paper/2307.09259/) (2023)
- [The Vietoris-Rips complexes of a circle](https://mlsanity.com/paper/1503.03669/) (2015)
- [K"unneth Formulae in Persistent Homology](https://mlsanity.com/paper/1910.05656/) (2019)

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

- [From Fourier Amplitudes to Persistence Images for Impact-Echo Defect Detection](https://mlsanity.com/paper/2610.18732/) (2026)

## Abstract (arXiv)

To analyze the topological properties of the given discrete data, one needs to consider a continuous transform called filtration. Persistent homology serves as a tool to track changes of homology in the filtration. The outcome of the topological analysis of data varies depending on the choice of filtration, making the selection of filtration crucial. Filtration learning is an attempt to find an optimal filtration that minimizes the loss function. Exact Multi-parameter Persistent Homology (EMPH) has been recently proposed, particularly for topological time-series analysis, that utilizes the exact formula of rank invariant instead of calculating it. In this paper, we propose a framework for filtration learning of EMPH. We formulate an optimization problem and propose an algorithm for solving the problem. We then apply the proposed algorithm to several classification problems. Particularly, we derive the exact formula of the gradient of the loss function with respect to the filtration parameters, which makes it possible to directly update the filtration without using automatic differentiation, significantly enhancing the learning process.