It has been known that the curvature of data spaces plays a role in data analysis. For example, the Frechet mean (intrinsic mean) always exists uniquely for a probability measure on a non-positively curved metric space. In this paper, we use the curvature of data spaces in a novel manner. A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. The p...

We remark that the result presented here can be proved more simply using the closed graph theorem. However we believe that our results can be used to prove a more interesting result. Details to follow in a later paper. One of the remarkable features of the space of holomorphic functions (in either one or several variables) is that the standard Frechet space topologies— say, for example, the L o...

Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that controls vanish on large parts underlying domain. Typical candidates for those integral Lebesgue spaces based $\ell_p$-metric $p\in[0,1)$ which nonconvex as well non-Lipschitz and, thus, variationally challenging. In this paper, we derive exact formulas Frechet, limiting, a...

Received (received date) Revised (revised date) Communicated by (Name) The article analyzes similarity of closed polygonal curves in Frechet metric, which is stronger than the well-known Hausdorff metric and therefore is more appropriate in some applications. An algorithm that determines whether the Frechet distance between two closed polygonal curves with m and n vertices is less than a given ...