Harmonic mappings and distance function

We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with $C^{1,\alpha}$ ($\alpha<1$), respectively $C^{1,1}$ compact boundary is bi-Lipschitz. The distance function with respect to the boundary of the image domain is used. This in turn extends a similar result of the author in \cite{kalajan} for Jordan domains, where stronger boundary conditions for the image domain were needed.Comment: 10 pages, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienz

The Heston Riemannian distance function

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and the sub-Riemannian distance function in the Grushin plane. For the Grushin plane, we establish an explicit formula for the Legendre-Fenchel transform of the limiting cumulant generating function and prove a partial large deviation principle that is true only inside a special set

Geodesic Distance Function Learning via Heat Flow on Vector Fields

Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if the original manifold is not Euclidean. Note that the distance function on a manifold can always be well-defined. In this paper, we propose to learn the distance function directly on the manifold without embedding. We first provide a theoretical characterization of the distance function by its gradient field. Based on our theoretical analysis, we propose to first learn the gradient field of the distance function and then learn the distance function itself. Specifically, we set the gradient field of a local distance function as an initial vector field. Then we transport it to the whole manifold via heat flow on vector fields. Finally, the geodesic distance function can be obtained by requiring its gradient field to be close to the normalized vector field. Experimental results on both synthetic and real data demonstrate the effectiveness of our proposed algorithm