Non-isotropic Persistent Homology: Leveraging the Metric Dependency of
  PH
Vincent P. Grande Michael T. Schaub
Persistent Homology is a widely used topological data analysis tool that
creates a concise description of the topological properties of a point cloud
based on a specified filtration. Most filtrations used for persistent homology
depend (implicitly) on a chosen metric, which is typically agnostically chosen
as the standard Euclidean metric on $\mathbb{R}^n$. Recent work has tried to
uncover the 'true' metric on the point cloud using distance-to-measure
functions, in order to obtain more meaningful persistent homology results. Here
we propose an alternative look at this problem: we posit that information on
the point cloud is lost when restricting persistent homology to a single
(correct) distance function. Instead, we show how by varying the distance
function on the underlying space and analysing the corresponding shifts in the
persistence diagrams, we can extract additional topological and geometrical
information. Finally, we numerically show that non-isotropic persistent
homology can extract information on orientation, orientational variance, and
scaling of randomly generated point clouds with good accuracy and conduct some
experiments on real-world data.