# A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions

arXiv: https://arxiv.org/abs/2306.11170 · HTML: https://mlsanity.com/paper/2306.11170/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 65 references that are in the map.
- First posted on arXiv 19 Jun 2023.

## Builds on (most cited first, 19 of 19)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Learning metrics for persistence-based summaries and applications for graph classification](https://mlsanity.com/paper/1904.12189/) (2019)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [RetGK: Graph Kernels based on Return Probabilities of Random Walks](https://mlsanity.com/paper/1809.02670/) (2018)
- [An Introduction to Multiparameter Persistence](https://mlsanity.com/paper/2203.14289/) (2022)
- [A Kernel for Multi-Parameter Persistent Homology](https://mlsanity.com/paper/1809.10231/) (2018)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [Algebraic Stability of Zigzag Persistence Modules](https://mlsanity.com/paper/1604.00655/) (2016)
- [Generalized Persistence Diagrams for Persistence Modules over Posets](https://mlsanity.com/paper/1810.11517/) (2018)
- [Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions](https://mlsanity.com/paper/2107.06800/) (2021)
- [Stability of 2-Parameter Persistent Homology](https://mlsanity.com/paper/2010.09628/) (2020)
- [The rank invariant stability via interleavings](https://mlsanity.com/paper/1412.3374/) (2014)
- [Multi-parameter Module Approximation: an efficient and interpretable invariant for multi-parameter persistence modules with guarantees](https://mlsanity.com/paper/2206.02026/) (2022)
- [Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules](https://mlsanity.com/paper/1904.03766/) (2019)
- [Elder-Rule-Staircodes for Augmented Metric Spaces](https://mlsanity.com/paper/2003.04523/) (2020)
- [Smart Vectorizations for Single and Multiparameter Persistence](https://mlsanity.com/paper/2104.04787/) (2021)

## Built on by (most cited first, 7 of 7)

- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)
- [Nonparametric regression on random geometric graphs sampled from submanifolds](https://mlsanity.com/paper/2405.20909/) (2024)
- [PROXI: Challenging the GNNs for Link Prediction](https://mlsanity.com/paper/2410.01802/) (2024)
- [CuMPerLay: Learning Cubical Multiparameter Persistence Vectorizations](https://mlsanity.com/paper/2510.12795/) (2025)
- [TopoPrune: Robust Data Pruning via Unified Latent Space Topology](https://mlsanity.com/paper/2602.02739/) (2026)
- [COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules](https://mlsanity.com/paper/2609.22012/) (2026)
- [The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology](https://mlsanity.com/paper/2610.05441/) (2026)

## Abstract (arXiv)

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is em persistent homology, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. In particular, a central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on em decompositions of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.