# A Stable Multi-Scale Kernel for Topological Machine Learning

arXiv: https://arxiv.org/abs/1412.6821 · HTML: https://mlsanity.com/paper/1412.6821/

## Key facts

- Cited by 74 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 21 Dec 2014.

## Built on by (most cited first, 30 of 74)

- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Cross-Scale MAE: A Tale of Multi-Scale Exploitation in Remote Sensing](https://mlsanity.com/paper/2401.15855/) (2024)
- [Link Prediction with Persistent Homology: An Interactive View](https://mlsanity.com/paper/2102.10255/) (2021)
- [Topological Detection of Trojaned Neural Networks](https://mlsanity.com/paper/2106.06469/) (2021)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [RipsNet: a general architecture for fast and robust estimation of the persistent homology of point clouds](https://mlsanity.com/paper/2202.01725/) (2022)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [Neural Approximation of Graph Topological Features](https://mlsanity.com/paper/2201.12032/) (2022)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [Topological Attention for Time Series Forecasting](https://mlsanity.com/paper/2107.09031/) (2021)
- [TopoTxR: A Topological Biomarker for Predicting Treatment Response in Breast Cancer](https://mlsanity.com/paper/2105.06049/) (2021)
- [Tensor-view Topological Graph Neural Network](https://mlsanity.com/paper/2401.12007/) (2024)
- [Topological Signatures vs. Gradient Histograms: A Comparative Study for Medical Image Classification](https://mlsanity.com/paper/2507.03006/) (2025)
- [Estimation and Quantization of Expected Persistence Diagrams](https://mlsanity.com/paper/2105.04852/) (2021)
- [Graph Filtration Kernels](https://mlsanity.com/paper/2110.11862/) (2021)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Learning Hyperbolic Representations of Topological Features](https://mlsanity.com/paper/2103.09273/) (2021)
- [A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science](https://mlsanity.com/paper/2207.10552/) (2022)
- [Augmenting Molecular Deep Generative Models with Topological Data Analysis Representations](https://mlsanity.com/paper/2106.04464/) (2021)
- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [D-GRIL: End-to-End Topological Learning with 2-parameter Persistence](https://mlsanity.com/paper/2406.07100/) (2024)
- [Topology Applied to Machine Learning: From Global to Local](https://mlsanity.com/paper/2103.05796/) (2021)
- [A computationally efficient framework for vector representation of persistence diagrams](https://mlsanity.com/paper/2109.08239/) (2021)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [Landslide Topology Uncovers Failure Movements](https://mlsanity.com/paper/2310.09631/) (2023)
- [SIFM: A Foundation Model for Multi-granularity Arctic Sea Ice Forecasting](https://mlsanity.com/paper/2410.14732/) (2024)
- [Topological Information Retrieval with Dilation-Invariant Bottleneck Comparative Measures](https://mlsanity.com/paper/2104.01672/) (2021)

## Abstract (arXiv)

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.