# Convolutional Persistence Transforms

arXiv: https://arxiv.org/abs/2208.02107 · HTML: https://mlsanity.com/paper/2208.02107/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 75 references that are in the map.
- First posted on arXiv 3 Aug 2022.

## Builds on (most cited first, 25 of 25)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm](https://mlsanity.com/paper/2002.03293/) (2020)
- [giotto-tda: A Topological Data Analysis Toolkit for Machine Learning and Data Exploration](https://mlsanity.com/paper/2004.02551/) (2020)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [An Introduction to Multiparameter Persistence](https://mlsanity.com/paper/2203.14289/) (2022)
- [A Fast and Robust Method for Global Topological Functional Optimization](https://mlsanity.com/paper/2009.08496/) (2020)
- [Continuation of Point Clouds via Persistence Diagrams](https://mlsanity.com/paper/1506.03147/) (2015)
- [Embeddings of Persistence Diagrams into Hilbert Spaces](https://mlsanity.com/paper/1905.05604/) (2019)
- [The Weighted Euler Curve Transform for Shape and Image Analysis](https://mlsanity.com/paper/2004.11128/) (2020)
- [From Geometry to Topology: Inverse Theorems for Distributed Persistence](https://mlsanity.com/paper/2101.12288/) (2021)
- [Comparing persistence diagrams through complex vectors](https://mlsanity.com/paper/1505.01335/) (2015)
- [Inverse Problems in Topological Persistence](https://mlsanity.com/paper/1810.10813/) (2018)
- [Reconstructing Embedded Graphs from Persistence Diagrams](https://mlsanity.com/paper/1912.08913/) (2019)
- [Rethinking Persistent Homology for Visual Recognition](https://mlsanity.com/paper/2207.04220/) (2022)
- [Stabilizing the unstable output of persistent homology computations](https://mlsanity.com/paper/1512.01700/) (2015)
- [Nonembeddability of Persistence Diagrams with p>2 Wasserstein Metric](https://mlsanity.com/paper/1910.13935/) (2019)
- [Topological analysis of scalar fields with outliers](https://mlsanity.com/paper/1412.1680/) (2014)
- [A Multi-parameter Persistence Framework for Mathematical Morphology](https://mlsanity.com/paper/2103.13013/) (2021)
- [Learning Simplicial Complexes from Persistence Diagrams](https://mlsanity.com/paper/1805.10716/) (2018)
- [Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes](https://mlsanity.com/paper/2110.14676/) (2021)

## Abstract (arXiv)

In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution filter as a local motif, the persistence diagram of the resulting convolution describes the way the motif is distributed across the simplicial complex. This pipeline, which we call convolutional persistence, extends the capacity of topology to observe patterns in such data. Moreover, we prove that (generically speaking) for any two labeled complexes one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams is an injective invariant. This is proven by showing convolutional persistence to be a special case of another topological invariant, the Persistent Homology Transform. Other advantages of convolutional persistence are improved stability, greater flexibility for data-dependent vectorizations, and reduced computational complexity for certain data types. Additionally, we have a suite of experiments showing that convolutions greatly improve the predictive power of persistence on a host of classification tasks, even if one uses random filters and vectorizes the resulting diagrams by recording only their total persistences.