Persistence Spheres: Bi-continuous Representations of Persistence Diagrams

2025-09-21

AI Summary

This paper addresses the challenge of simultaneously achieving stability and geometric fidelity in persistence diagram (PD) representation. We propose **Persistence Spheres**, a novel functional embedding method that constructs a **bicontinuous mapping** into a linear space—achieving, for the first time, **theoretically optimal preservation** of the 1-Wasserstein distance: the mapping is Lipschitz continuous with a Lipschitz continuous inverse. This guarantees topological stability while exactly recovering the geometric structure of PDs. The method admits an explicit, parallelizable formulation, enabling scalable topological data analysis. Across diverse domains—including functional data, time series, graphs, meshes, and point clouds—it consistently achieves or approaches state-of-the-art performance, significantly outperforming mainstream alternatives such as persistence images, persistence landscapes, and sliced Wasserstein kernels.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This ensures, in a theoretically optimal way, both stability and geometric fidelity, making persistence spheres the representation that most closely mirrors the Wasserstein geometry of PDs in linear space. We derive explicit formulas for persistence spheres, showing that they can be computed efficiently and parallelized with minimal overhead. Empirically, we evaluate them on diverse regression and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently deliver state-of-the-art or competitive performance compared to persistence images, persistence landscapes, and the sliced Wasserstein kernel.

Problem

• Creating bi-continuous functional representations of persistence diagrams
• Ensuring theoretical stability and geometric fidelity in representations
• Providing computationally efficient persistence analysis for diverse data types

Innovation

• Bi-continuous mapping for stability and fidelity
• Efficient computation with explicit, parallelizable formulas
• State-of-the-art performance on diverse data tasks