Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates

• URL: http://arxiv.org/abs/2402.01493v2
• Date: Wed, 15 May 2024 09:17:13 GMT
• Title: Sliced-Wasserstein Estimation with Spherical Harmonics as Control Variates
• Authors: Rémi Leluc, Aymeric Dieuleveut, François Portier, Johan Segers, Aigerim Zhuman,
• Abstract summary: Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein resulting for associated one-dimensional projections. Spherical harmonics are distances on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. An improved rate of convergence, compared to Monte Carlo, is established for general measures.
• Score: 17.237390976128097
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Sliced-Wasserstein (SW) distance between probability measures is defined as the average of the Wasserstein distances resulting for the associated one-dimensional projections. As a consequence, the SW distance can be written as an integral with respect to the uniform measure on the sphere and the Monte Carlo framework can be employed for calculating the SW distance. Spherical harmonics are polynomials on the sphere that form an orthonormal basis of the set of square-integrable functions on the sphere. Putting these two facts together, a new Monte Carlo method, hereby referred to as Spherical Harmonics Control Variates (SHCV), is proposed for approximating the SW distance using spherical harmonics as control variates. The resulting approach is shown to have good theoretical properties, e.g., a no-error property for Gaussian measures under a certain form of linear dependency between the variables. Moreover, an improved rate of convergence, compared to Monte Carlo, is established for general measures. The convergence analysis relies on the Lipschitz property associated to the SW integrand. Several numerical experiments demonstrate the superior performance of SHCV against state-of-the-art methods for SW distance computation.

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