Free probability via entropic optimal transport (2309.12196v3)

Abstract: Let $\mu$ and $\nu$ be probability measures on $\mathbb{R}$ with compact support, and let $\mu \boxplus \nu$ denote their additive free convolution. We show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of $\mu$ and $\nu$, we have \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) = \sup_{\Pi} \left{ \mathbf{E}\Pi[\log(z - (X+Y)] - H(\Pi|\mu \otimes \nu) \right}, \end{equation*} where the supremum is taken over all couplings $\Pi$ of the probability measures $\mu$ and $\nu$, and $H(\Pi|\mu \otimes \nu)$ denotes the relative entropy of a coupling $\Pi$ against product measure. We prove similar formulas for the multiplicative free convolution $\mu \boxtimes \nu$ and the free compression $[\mu]\tau$ of probability measures, as well as for multivariate free operations. Thus the integrals of a log-potential against the fundamental measure operations of free probability may be formulated in terms of entropic optimal transport problems. The optimal couplings in these variational descriptions of the free probability operations can be computed explicitly, and from these we can then deduce the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution. We use our optimal transport formulations to derive new inequalities relating free and classical operations on probability measures, such as the inequality \begin{equation*} \int_{-\infty}^\infty \log(z - x) \mu \boxplus \nu (\mathrm{d}x) \geq \int_{-\infty}^{\infty} \log(z-x) \mu \ast \nu( \mathrm{d}x) \end{equation*} relating free and classical convolution. Our approach is based on applying a large deviation principle on the symmetric group to the quadrature formulas of Marcus, Spielman and Srivastava.

• The paper introduces an entropic optimal transport framework that reformulates free convolutions using variational principles.
• The paper develops explicit optimal couplings that bridge classical R- and S-transform methods with free probability operations.
• The paper reveals new inequalities and a large deviation approach that clarify spectral properties in random matrix theory.

An Analysis of "Free Probability via Entropic Optimal Transport"

The paper by Arizmendi and Johnston titled "Free Probability via Entropic Optimal Transport" advances the field of free probability by establishing connections with entropic optimal transport. The authors address foundational aspects of free probability operations such as additive free convolution (denoted as ⊞), multiplicative free convolution (⊠), and free compression (denoted by ⊠τ). They apply entropic optimal transport theory to derive variational formulas for these operations, which are traditionally characterized through algebraic tools like R-transform and S-transform or combinatorial methods related to free cumulants.

Key Contributions

1. Entropic Optimal Transport Framework: The paper extends the entropic optimal transport framework to free probability, providing a new perspective on convolution operations. The authors demonstrate that the fundamental measure operations of free probability can be understood and derived via variational problems associated with entropic optimal transport.
2. Formulation and Optimization: For free convolutions, the authors introduce entropic optimal transport problems where the objective is to maximize the expected value of a log-potential function minus a relative entropy term. The relative entropy is calculated against the product measure of the involved probability measures. They successfully compute explicit optimal couplings for these variational problems, leading to the classical R- and S-transform descriptions of free convolutions.
3. Inequalities and Relations with Classical Probability: The paper provides new inequalities relating free and classical convolutions, emphasizing that free operations induce measures with different tail behaviors relative to their classical counterparts.
4. Large Deviation Principle: The authors incorporate a large deviation approach, particularly employing the quadrature formulas of Marcus, Spielman, and Srivastava. This principle plays a crucial role in analyzing asymptotic behaviors and structural properties of spectra from random matrices, giving a more refined understanding of the free probability operations.
5. Multivariate and Complexity Extensions: The paper generalizes its formulations to multivariate settings and further entertains complexities such as free compression, demonstrating the robustness and applicability of the proposed framework across a range of probability measures.

Numerical Insights and Theoretical Implications

The authors successfully link the entropic optimal transport formulations to numerically significant features of free probability, such as transformations of Cauchy transforms and their roles in defining free convolutions. The findings highlight the potential reduction in the complexity of analyzing spectral distributions of large random matrices via entropic transport theory, proposing a path forward in theoretical studies of random matrices.

Speculation on Future Developments in AI

While the current paper offers a mathematical expansion rather than directly addressing artificial intelligence, its implications for matrix operations might influence AI fields that utilize large computational frameworks, such as deep learning and generative models. The ability to accurately and efficiently analyze spectra of matrix convolutions can enhance the training and inference phases in neural networks, particularly in optimization and model robustness.

Conclusion

Arizmendi and Johnston’s paper provides a rigorous and comprehensive treatment of free probability through an entropic optimal transport lens. By bridging these advanced mathematical concepts, it paves the way for further interdisciplinary explorations and strengthens theoretical underpinnings that could have applications across various scientific domains.