[Paper Review] Sliced Wasserstein Kernel for Persistence Diagrams

Mathieu Carrière, Marco Cuturi+1 more|arXiv (Cornell University)|Jun 11, 2017

Topological and Geometric Data AnalysisComputer Science33 references87 citations

TL;DR

Introduces a Sliced Wasserstein kernel for persistence diagrams that is provably stable and discriminative, along with an efficient approximation, and shows superior classification performance on benchmarks compared to prior kernels.

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.

Motivation & Objective

• Motivate the use of persistence diagrams in learning and address the lack of a Hilbert space structure for standard diagrams.
• Propose a kernel based on the Sliced Wasserstein distance to inject persistence diagrams into RKHS with stability and discriminativity.
• Establish theoretical guarantees relating SW to diagram distance d1 and demonstrate practical computation methods.
• Compare the proposed kernel to existing kernels on benchmark classification tasks and show improved performance.

Proposed method

• Define the Sliced Wasserstein (SW) distance by projecting diagrams onto lines through the origin, integrating optimal transport on these projections over all directions.
• Show that SW is conditionally negative definite and thus define a positive definite kernel k_SW(x,y)=exp(-SW(x,y)/(2 sigma^2)).
• Prove stability: SW is bounded by a multiple of d1 between diagrams.
• Prove discriminativity: SW bounds d1 from below up to a factor for bounded-degree diagrams (bounded cardinality).
• Provide an efficient approximation algorithm that samples directions to compute SW in O(M N log N) time, and an exact variant for bounded cardinalities.
• Demonstrate injectivity of the SW feature map on finite, bounded diagrams and discuss universality via exponentiation.

Experimental results

Research questions

• RQ1 Can a kernel for persistence diagrams be both stable and discriminative with respect to diagram distances?
• RQ2 Does the Sliced Wasserstein distance provide a practical, computable surrogate that preserves the geometry of persistence diagrams when used in kernel methods?
• RQ3 How does k_SW compare to existing kernels (PSS, PWG) in classification tasks and computational cost?
• RQ4 Is there an efficient approximation scheme that maintains kernel properties while speeding up computation?

Key findings

• SW is a stable and discriminative kernel for persistence diagrams with respect to the first diagram distance d1.
• SW is strongly equivalent in its induced RKHS distance to d1 up to monotone, continuous distortions, enabling geometry-preserving embeddings.
• An efficient approximation (SW_M) using M sampled directions provides competitive accuracy with reduced computation time.
• Empirical benchmarks show k_SW outperforms k_PSS and k_PWG on several tasks (Orbit, Texture, Human, Airplane, Ant, Bird, FourLeg, Octopus, Fish).
• The paper provides exact and approximate computation schemes, with exact O(N^2 log N) time for general-position diagrams and O(M N log N) for the approximation.
• Experiments indicate that a small number of directions suffices for good classification performance.