An approximation of the squared Wasserstein distance and an application to Hamilton-Jacobi equations

Abstract: We provide a simple $C^{1,1}$ approximation of the squared Wasserstein distance on R^d when one of the two measures is fixed. This approximation converges locally uniformly. More importantly, at points where the differential of the squared Wasserstein distance exists, it attracts the differentials of the approximations at nearby points. Our method relies on the Hilbertian lifting of PL Lions and on the regularization in Hilbert spaces of Lasry and Lions. We then provide an application of this result by using it to establish a comparison principle for an Hamilton-Jacobi equation on the set of probability measures.

• The paper introduces a C¹,¹ approximation of the squared Wasserstein distance using sup-convolution regularization and Hilbertian lifting.
• It establishes differentiability properties that effectively integrate the Wasserstein distance in solving Hamilton-Jacobi equations on probability measures.
• The findings create a robust framework for addressing non-smooth optimization problems and pave the way for future computational and theoretical research.

An Approximation of the Squared Wasserstein Distance and its Application to Hamilton-Jacobi Equations

The research paper by Charles Bertucci and Pierre-Louis Lions presents a nuanced approach to approximating the squared Wasserstein distance and explores its implications for Hamilton-Jacobi equations in the space of probability measures. This study provides significant insights into addressing the differentiability issues of the Wasserstein distance and illustrates the mathematical sophistication required to approximate this measure effectively.

Summary of the Approach

The main focus of the paper is the approximation of the squared Wasserstein distance, which traditionally poses challenges due to its non-smooth nature. By leveraging the Hilbertian lifting technique pioneered by Lions and the regularization strategy in Hilbert spaces as developed by Lasry and Lions, the authors construct a ${C^{1,1}}$ approximation. This is a pivotal advancement as it ensures that the approximation converges locally uniformly and aligns with the differentiability properties of the squared Wasserstein distance.

The authors articulate a sophisticated methodology that involves a sup-convolution regularization technique to achieve this approximation, demonstrating a clear path toward solving optimization problems where the Wasserstein distance plays a critical role.

Implications for Hamilton-Jacobi Equations

The paper extends the implications of this approximation by exploring its application to Hamilton-Jacobi equations on the set of probability measures. It establishes a comparison principle for these equations, which traditionally confront many analytical challenges, especially when conventional methods fail due to the non-smooth nature of critical terms like the Wasserstein distance.

Through an intricate argument involving viscosity solutions and the introduction of the entropy functional, the study avoids the pitfalls associated with the lack of smoothness, ultimately enabling the use of the Wasserstein distance as a test function in this context. The analysis further proposes using the approximation to handle singular terms in Hamilton-Jacobi equations, particularly those involving divergence.

Numerical Results and Theoretical Implications

The paper does not center explicitly on numerical evaluations, but the mathematical results hold strong implications for computational applications. The authors assert that their approach not only manages the approximation of the Wasserstein distance effectively but does so while maintaining mathematical rigor.

The paper's contributions underline the feasibility of using viscosity solutions with Hamilton-Jacobi equations in complex probabilistic spaces. Moreover, the regularity results obtained serve as bedrocks for further investigation into more expansive models or potential integration with machine learning methodologies, particularly in areas requiring robust distance metrics.

Future Directions

While the study provides key insights into both approximations of distance measures and their applications, it paves the way for several future research paths. Potential areas of exploration include:

• Extending the regularization techniques to other forms of distance metrics within the probability measures framework.
• Exploring the implementation of these mathematical techniques in real-world applications, such as stochastic control or gradient-driven decision processes.
• Investigating the intersection of these theoretical advancements with data-driven approaches in fields like adversarial networks or probabilistic graphical models.

In summary, this paper significantly contributes to the ongoing discourse on the interplay between differential equations, probability measures, and functional analysis. It offers a robust framework that experienced researchers can further expand to explore both theoretical and practical applications in various domains, bridging gaps between abstract theoretical constructs and computational pragmatism.