Abstract:
In the last decade, there has been increasing interest in topological
data analysis, a new methodology for using geometric structures in
data for inference and learning. A central theme in the area is the
idea of persistence, which in its most basic form studies how measures
of shape change as a scale parameter varies. There are now a number of
frameworks that support statistics and machine learning in this
context. However, in many applications there are several different
parameters one might wish to vary: for example, scale and density. In
contrast to the one-parameter setting, techniques for applying
statistics and machine learning in the setting of multiparameter
persistence are not well understood due to the lack of a concise
representation of the results.
We introduce a new descriptor for multiparameter persistence, which we
call the Multiparameter Persistence Image, that is suitable for
machine learning and statistical frameworks, is robust to
perturbations in the data, has finer resolution than existing
descriptors based on slicing, and can be efficiently computed on data
sets of realistic size. Moreover, we demonstrate its efficacy by
comparing its performance to other multiparameter descriptors on
several classification tasks.
