When considering high-dimensional objects, the Euclidean distance is not always the most suitable for assessing the actual distance between data points. It is of interest to study conformal deformations of the Euclidean metric, using functions that take the distribution of the data into account to provide a metric that represents better the geometry and statistical properties of this data.
The Fermat distance is an example of such metric that deforms space by bringing points closer together in areas of high density. However its theoretical analysis is complex and it is by definition restricted to measures with density.
In this talk I will discuss these limitations and introduce a variant of the Fermat distance defined for any measure that possesses strong stability and estimation properties.
I will also discuss conformal metrics in general and their regularity in order to provide estimation results.