Colloquium, Dept. of Mathematics and Applied Math. -- Γενικό Σεμινάριο

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Colloquium

Statistical optics of complex systems: From beam propagation through atmospheric turbulence to thermalization and thermodynamics

Abstract: We present a statistical-optics view of light in complex media, ranging from structured beam transport through atmospheric turbulence to thermodynamics and thermalization of weakly nonlinear multimode optical systems. The complexity arises either from the fluctuations of the refractive index in atmospheric turbulence or from the combination of nonlinearity with the presence of a large number of modes in optical thermodynamics.

Abstract: Generative Adversarial Networks (GANs) represent one of the most influential families of generative models. This talk will introduce the core principles of GANs and their formulation as divergence minimization problems. By employing variational (duality-based) representations of divergences, the intractable optimization problem can be reframed as a tractable two-player zero-sum game. While GANs have achieved remarkable success, training instability and convergence issues remain major challenges. To address these, we propose a general framework for constructing tailored divergences that bridge $f$-divergences and integral probability metrics, combining the stability properties of both. I will highlight key results demonstrating improved convergence and performance in challenging scenarios, such as heavy-tailed distributions and image generation. Finally, I will discuss a novel variant of GANs that naturally gives rise to Generative Gradient Flows and a corresponding Generative Particle Algorithm.

Abstract: The Shannon-Stam inequality, also known as Entropy Power Inequality (EPI), is a celebrated result in information theory, which goes back to Shannon's seminal paper. Nevertheless, the equality case has only been characterized under finite second moment or additional regularity assumptions. In this talk, we show that there is equality in the Shannon-Stam inequality if and only if the random variables in question are Gaussian, assuming nothing else than finiteness of differential entropies.

To that end, we present necessary and sufficient conditions for the validity of de Bruijn's identity. The latter links the derivative of entropy under Gaussian perturbations to Fisher information. While it has been widely applied and rigorously derived under mild moment or regularity assumptions, the necessary and sufficient conditions ensuring the validity of the required exchanges of differentiation and integration have not been fully identified in the literature.

Finally, we discuss the question of stability in the Shannon-Stam inequality, motivated by connections with recent advances in convex geometry and additive combinatorics. After reviewing some existing negative results, we conclude with a qualitative stability theorem in the weak convergence sense, valid under very mild assumptions.