From genomics to transfer learning

Today there are a number of interesting talks but they are not what I am familiar with. I heard many new concepts for the first time.

Gabriel Peyré: Ground metric learning with applications in genomics Regulaised Wasserstein metric: cost function plus some regulating functions in the metric Gromov-Wasserstein metric: distance of measures on different spaces Wasserstein singular vectors: some kind of eigenvectors with respect to Wasserstein transfer operator

Jan Maas Anisotropy and the infinitesimal model (Blackboard talk) Talagrand’s inequality If $\mu$ satisfies a log-Sobolev inequalty $\operatorname{Ent}(\nu | \mu) \le \frac{1}{2K} I_2(\nu,\mu)$ with $I_2(\nu,\mu)=| \nabla \log (\frac{d \nu}{d \mu})|$ then $W_2(\mu,\nu) \le \sqrt{\frac{1}{2K} \operatorname{Ent}(\nu | \mu)}$

If $\mu$ is $K-\log$ concave with density $e^{-U}, D^2 U \ge K I$, $\nu = e^{-H} \mu$, $H$ $L$-Lipschtiz. Then $W_{\infty}(\mu,\nu)\le \frac{L}{K}$ Proof: Using Langevin dynamics How does the Langevin dynamics $(X_t,Y_t)$ constructed with $U$ and $H$ with a Brownian motion,

$| X_t - Y_t | \le \frac{L}{K}$ implies the $W_\infty (\mu ,\nu) \le \frac{L}{K}$?

Similar results for Fisher’s infinitesimal model was derived.

Laetitia Chapel: Fast Optimal Transport through Sliced Generalized Wasserstein Geodesics Approximation of Wasserstein distance using sliced Wasserstein (SW) distance, SWGG(SW-generalised geodesic) which are upper bounds and lower bounds of Wasserstein. They are easier for numerical calculation even for high dimension.

Massimiliano Pontil: Transfer Operator Learning Related to transfer learning

Guillaume Carlier: Displacement Smoothness of Entropic Optimal Transport and Applications to some Evolution Equations and Systems Perturb Wasserstein distance by adding a relative entropy to the usual cost function. Sinkhorn iteration: Become a fix point problem to show convergence in this entropic metric. Application to Gradient flow: well-posed of GF using Sinkhorn divergence $\rho \mapsto E(\rho,\mu)-frac{1}{2}E(\rho,\rho)-\frac{1}{2}E(\mu,\mu)$ semi-geostropic flow in meteorology.