Wasserstein Stability for Persistence Diagrams (2006.16824v5)

Abstract: The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of perturbations. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this paper, we provide new stability results with respect to the $p$-Wasserstein distance between persistence diagrams. This includes an elementary proof for the setting of functions on sufficiently finite spaces in terms of the $p$-norm of the perturbations, along with an algebraic framework for $p$-Wasserstein distance which extends the results to wider class of modules. We also provide apply the results to a wide range of applications in topological data analysis (TDA) including topological summaries, persistence transforms and the special but important case of Vietoris-Rips complexes.

• The paper introduces p-Wasserstein stability as a robust alternative to the bottleneck distance for persistence diagrams.
• The authors develop an algebraic framework extending stability results to a wider class of persistence modules.
• Applications in image analysis and other TDA methods demonstrate practical benefits and improved resilience to outliers.

Wasserstein Stability for Persistence Diagrams

In the paper "Wasserstein Stability for Persistence Diagrams" by Primoz Skraba and Katharine Turner, the authors offer new insights into the stability of persistence diagrams within the field of computational topology. Traditionally, stability results for persistence diagrams have emphasized the bottleneck distance, utilizing the $\infty$-norm. However, Skraba and Turner pivot towards using the $p$-Wasserstein distance, providing stability results that are more robust against outliers and better suited to practical applications, notably in applied topology and data analysis.

Main Contributions

1. Introduction of $p$-Wasserstein Stability: The paper presents elementary proofs of stability using the $p$-Wasserstein distance for persistence diagrams, particularly for sublevel set filtrations of functions defined over finite cellular complexes. The authors establish an algebraic framework extending these results to a broader class of modules. This approach alleviates the issues associated with the traditional bottleneck distance, such as its sensitivity to outliers.
2. Novel Algebraic Framework: Extending beyond the finite setting, Skraba and Turner develop an algebraic framework to describe $p$-Wasserstein distance for persistence modules. They introduce an alternate intermediate object—a span of persistence modules—that serves as an interpolating entity in proving algebraic stability. This framework is crucial for proving stability in contexts where classical persistence diagrams do not apply.
3. Applications in Topological Data Analysis: The paper outlines several applications of this theoretical advancement, including image analysis, persistent homology transforms, and Vietoris-Rips filtrations. For example, in the analysis of grayscale images, a cubical complex is associated with the pixel values, and the authors demonstrate stability results that factor in the $p$-Wasserstein distance. This offers practical relevance for analyzing and comparing high-dimensional data.

Implications and Future Directions

The authors' work opens the door to new possibilities in applied topology by mitigating the bottleneck distance's limitations, paving the way for more adaptable and precise methods in topological data analysis. The introduction of algebraic stability encourages consideration of non-standard persistence settings, such as multiparameter and sheaf-based persistence, which could significantly enhance the flexibility and applicability of persistent homology methods.

Furthermore, the paper suggests that traditional bottleneck results, while foundational, are sometimes overly pessimistic, particularly in scenarios heavily influenced by outliers. By focusing instead on $p$-Wasserstein stability in both cellular and algebraic contexts, Skraba and Turner provide a foundation for more refined analytical techniques. This advancement has potential implications for stochastic topology and could improve methodologies for modeling and understanding random phenomena within datasets.

Conclusion

Primoz Skraba and Katharine Turner's work is a substantial contribution to the field of computational topology and topological data analysis, emphasizing more realistic and computationally feasible measures for persistence diagrams through the $p$-Wasserstein distance. Theoretical developments, coupled with practical applications, underscore the importance of revisiting foundational stability frameworks to better accommodate a wider array of data variances and inherent complexities in modern computational scenarios. As the field evolves, one can expect further refinements and expansions of these concepts, leading to improved tools for data analysis and interpretation in various scientific domains.