# Statistical learning on measures: an application to persistence diagrams

arXiv: https://arxiv.org/abs/2303.08456 · HTML: https://mlsanity.com/paper/2303.08456/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 66 references that are in the map.
- First posted on arXiv 15 Mar 2023.

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

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (2017)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [The UEA multivariate time series classification archive, 2018](https://mlsanity.com/paper/1811.00075/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [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)
- [Multiple Instance Learning for Digital Pathology: A Review on the State-of-the-Art, Limitations & Future Potential](https://mlsanity.com/paper/2206.04425/) (2022)
- [Learning Theory for Distribution Regression](https://mlsanity.com/paper/1411.2066/) (2014)
- [Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations](https://mlsanity.com/paper/2008.09165/) (2020)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Distribution-Free Distribution Regression](https://mlsanity.com/paper/1302.0082/) (2013)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [Supervised learning of sheared distributions using linearized optimal transport](https://mlsanity.com/paper/2201.10590/) (2022)
- [A chain rule for the expected suprema of Gaussian processes](https://mlsanity.com/paper/1411.2635/) (2014)
- [Persistence Diagrams with Linear Machine Learning Models](https://mlsanity.com/paper/1706.10082/) (2017)
- [Structure and Stability of the 1-Dimensional Mapper](https://mlsanity.com/paper/1511.05823/) (2015)

## Built on by (most cited first, 3 of 3)

- [Discrete transforms of quantized persistence diagrams](https://mlsanity.com/paper/2312.17093/) (2023)
- [Hierarchical Clustering in LambdaCDM Cosmologies via Persistence Energy](https://mlsanity.com/paper/2401.01988/) (2024)
- [Exact Topological Compliance: Generating Persistence-Equivalent Graphs](https://mlsanity.com/paper/2610.18856/) (2026)

## Abstract (arXiv)

We consider a binary supervised learning classification problem where instead of having data in a finite-dimensional Euclidean space, we observe measures on a compact space X. Formally, we observe data DN = (μ1, Y1), ldots, (μN, YN) where μi is a measure on X and Yi is a label in 0, 1. Given a set F of base-classifiers on X, we build corresponding classifiers in the space of measures. We provide upper and lower bounds on the Rademacher complexity of this new class of classifiers that can be expressed simply in terms of corresponding quantities for the class F. If the measures μi are uniform over a finite set, this classification task boils down to a multi-instance learning problem. However, our approach allows more flexibility and diversity in the input data we can deal with. While such a framework has many possible applications, this work strongly emphasizes on classifying data via topological descriptors called persistence diagrams. These objects are discrete measures on ℝ2, where the coordinates of each point correspond to the range of scales at which a topological feature exists. We will present several classifiers on measures and show how they can heuristically and theoretically enable a good classification performance in various settings in the case of persistence diagrams.