# Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds

arXiv: https://arxiv.org/abs/2307.09259 · HTML: https://mlsanity.com/paper/2307.09259/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 42 references that are in the map.
- First posted on arXiv 18 Jul 2023.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Rethinking Network Design and Local Geometry in Point Cloud: A Simple Residual MLP Framework](https://mlsanity.com/paper/2202.07123/) (2022)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Kernel method for persistence diagrams via kernel embedding and weight factor](https://mlsanity.com/paper/1706.03472/) (2017)
- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Using persistent homology and dynamical distances to analyze protein binding](https://mlsanity.com/paper/1412.1394/) (2014)
- [PI-Net: A Deep Learning Approach to Extract Topological Persistence Images](https://mlsanity.com/paper/1906.01769/) (2019)
- [Persistent homology machine learning for fingerprint classification](https://mlsanity.com/paper/1711.09158/) (2017)
- [Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives](https://mlsanity.com/paper/2303.15385/) (2023)
- [Topological descriptors for 3D surface analysis](https://mlsanity.com/paper/1601.06057/) (2016)

## Built on by (most cited first, 8 of 8)

- [Topology-Aware 3D Gaussian Splatting: Leveraging Persistent Homology for Optimized Structural Integrity](https://mlsanity.com/paper/2412.16619/) (2024)
- [FOLIAGE: Towards Physical Intelligence World Models Via Unbounded Surface Evolution](https://mlsanity.com/paper/2506.03173/) (2025)
- [Filtration learning in exact multi-parameter persistent homology and classification of time-series data](https://mlsanity.com/paper/2406.19587/) (2024)
- [A Persistent Homology Design Space for 3D Point Cloud Deep Learning](https://mlsanity.com/paper/2604.04299/) (2026)
- [Conformable Convolution for Topologically Aware Learning of Complex Anatomical Structures](https://mlsanity.com/paper/2412.20608/) (2024)
- [Structure-First Point Cloud Learning with Mapper Region Graphs](https://mlsanity.com/paper/2602.05522/) (2026)
- [Non-negative Matrix Factorisation with Topological Regularisation](https://mlsanity.com/paper/2606.17531/) (2026)
- [Voronoi Histograms for Adaptive Vectorization of Expected Persistence Diagrams](https://mlsanity.com/paper/2607.27126/) (2026)

## Abstract (arXiv)

Machine learning for point clouds has been attracting much attention, with many applications in various fields, such as shape recognition and material science. For enhancing the accuracy of such machine learning methods, it is often effective to incorporate global topological features, which are typically extracted by persistent homology. In the calculation of persistent homology for a point cloud, we choose a filtration for the point cloud, an increasing sequence of spaces. Since the performance of machine learning methods combined with persistent homology is highly affected by the choice of a filtration, we need to tune it depending on data and tasks. In this paper, we propose a framework that learns a filtration adaptively with the use of neural networks. In order to make the resulting persistent homology isometry-invariant, we develop a neural network architecture with such invariance. Additionally, we show a theoretical result on a finite-dimensional approximation of filtration functions, which justifies the proposed network architecture. Experimental results demonstrated the efficacy of our framework in several classification tasks.