ToMAToMP: Robust and Multi-Parameter Topological Clustering

TL;DR

ToMAToMP is a novel topological clustering algorithm that handles multiple functions simultaneously, offering robustness, reduced parameter sensitivity, and improved clustering quality through multi-parameter persistent homology.

Contribution

It introduces ToMAToMP, the first multi-parameter topological clustering method with theoretical guarantees, addressing limitations of ToMATo such as sensitivity to outliers and single-function handling.

Findings

ToMAToMP outperforms baseline methods on various datasets.

The algorithm is robust to outliers and independent of graph tuning.

Numerical experiments demonstrate improved clustering quality.

Abstract

Topological clustering, and its main algorithm ToMATo, is a clustering method from Topological Data Analysis (TDA) which has been applied successfully in several applications during the last few years. This is due to its high versatility, as clusters are detected from the persistent components in the sublevel sets of any user-defined function (gene expression, pixel values, etc), and efficiency, as topological clustering enjoys robustness guarantees. However, ToMATo is also limited in several ways. First, a graph on the data points needs to be provided as a hyper-parameter of the method (whose fine-tuning is left to the user). Second, ToMATo is known to be very sensitive to outlier values in the function range. Finally, and most importantly, ToMATo can only handle one function at a time, whereas it is critical to use several functions in various applications. In this article, we introduce ToMAToMP: the first topological clustering method able to handle several functions at the same time with theoretical guarantees. More specifically, we leverage a recent tool from multi-parameter persistent homology, called MMA decomposition, to design our clustering algorithm, and prove that it enjoys robustness properties. As corollaries, we show that it can be used to make ToMATo independent of graph tuning, and robust to outliers. Finally, we provide a set of numerical experiments showcasing the efficiency and quality of the clusterings produced by ToMAToMP, by showing strong improvement over non-topological and topological baselines for various datasets.