Sliced Wasserstein Kernel for Persistence Diagrams (1706.03358v3)

Abstract: Persistence diagrams (PDs) play a key role in topological data analysis (TDA), in which they are routinely used to describe topological properties of complicated shapes. PDs enjoy strong stability properties and have proven their utility in various learning contexts. They do not, however, live in a space naturally endowed with a Hilbert structure and are usually compared with specific distances, such as the bottleneck distance. To incorporate PDs in a learning pipeline, several kernels have been proposed for PDs with a strong emphasis on the stability of the RKHS distance w.r.t. perturbations of the PDs. In this article, we use the Sliced Wasserstein approximation SW of the Wasserstein distance to define a new kernel for PDs, which is not only provably stable but also provably discriminative (depending on the number of points in the PDs) w.r.t. the Wasserstein distance $d_1$ between PDs. We also demonstrate its practicality, by developing an approximation technique to reduce kernel computation time, and show that our proposal compares favorably to existing kernels for PDs on several benchmarks.

• The paper presents a novel kernel that leverages the Sliced Wasserstein distance to embed persistence diagrams into a Hilbert space for conventional methods.
• The paper shows that the Sliced Wasserstein Kernel maintains robust stability while preserving discriminative capacity even under diagram perturbations.
• The paper introduces an efficient approximation method that reduces computational complexity to O(MN log N), outperforming traditional persistence diagram kernels in benchmarks.

Sliced Wasserstein Kernel for Persistence Diagrams: An Expert Overview

The paper "Sliced Wasserstein Kernel for Persistence Diagrams" presents a novel kernel for persistence diagrams, pivotal tools in Topological Data Analysis (TDA). Persistence diagrams are utilized to encapsulate the topological features of data, offering stability in the presence of noise or perturbations. Despite their robustness, the space of persistence diagrams lacks a Hilbert structure, posing challenges for integration into conventional machine learning workflows that require Hilbertian frameworks, such as Support Vector Machines (SVM) and Principal Component Analysis (PCA). This work leverages the Sliced Wasserstein distance to construct a kernel that promises not only stability but also maintains a discriminative capacity with respect to the first diagram distance.

Technical Contributions

The paper makes several technical contributions:

1. Novel Kernel Definition: The authors define a new Sliced Wasserstein Kernel (SWK) based on the Sliced Wasserstein distance, previously demonstrated as conditionally negative definite, thus making it suitable for kernel methods. This kernel is shown to preserve metric properties akin to the more traditional diagram distances, which are typically not negative semi-definite.
2. Stability and Discriminativity: SWK exhibits strong stability with a distortion bound directly related to the number of points in the persistence diagrams. More importantly, it presents a discriminative property, a desirable trait for practical machine learning applications, indicating that SWK's induced distance closely tracks the chosen diagram metric under perturbations.
3. Computational Efficiency: The authors propose an approximation algorithm that reduces the computational burden of evaluating the kernel, achieving time complexity proportional to $O(MN \log N)$, where $N$ is the cardinality bound on persistence diagram points, and $M$ is the number of directions sampled.
4. Compatibility with Existing Frameworks: By presenting their kernel in a Reproducing Kernel Hilbert Space (RKHS), the authors enable its integration into standard machine learning pipelines, bridging the gap between TDA and statistical learning.

Empirical Evaluation

The paper's empirical assessments focus on several benchmarks where persistence diagrams conventionally provide value, including tasks in orbit recognition, texture classification, and shape analysis. Across these tasks, SWK consistently outperformed existing kernels like the Persistence Scale Space (PSS) and Persistence Weighted Gaussian (PWG) kernels both in accuracy and computational efficiency. The results underline the ability of the SWK to effectively capture the underlying topological structures inherent to diverse data.

Implications and Future Directions

This work augments the TDA toolbox by offering a robust and efficient kernel method for persistence diagrams, potentially enhancing the deployment of topological methods in machine learning tasks. The immediate implications include improved classification performance and reduced computational costs in algorithms that utilize persistence diagrams as features.

Looking forward, there are several avenues for future research opened by this work:

• Extending to Other Distances: While this paper focuses on the first diagram distance, investigating extensions to other persistence diagram distances might enhance versatility.
• Adaptations in Deep Learning: Although traditional machine learning methods were the primary focus, exploring the utility of SWK in deep learning architectures could yield significant insights, particularly in convolutional neural networks where topological features might be critical.
• Parameter Tuning Strategies: Developing more sophisticated parameter tuning strategies beyond grid search could further optimize SWK’s integration into learning pipelines.

In conclusion, this paper enriches the intersection of computational topology and machine learning by providing a theoretically sound and empirically validated method for harnessing the power of persistence diagrams through a dedicated kernel, facilitating more seamless and effective use of TDA in various domains.