The persistence diagram is an increasingly useful tool arising from the field of Topological Data Analysis . However, using these diagrams in conjunction with machine learning techniques requires some mathematical finesse. The most success to date has come from finding methods for turning persistence diagrams into vectors in \mathbb{R in a way which preserves as much of the space of persistence diagrams as possible, commonly referred to as featurization. In this paper, we describe a mathematical framework for featurizing the persistence diagram space using template functions. These functions are general as they are only required to be continuous , have a compact support, and separate points . We discuss two example realizationsof these functions : tent functions and Chybeyshev interpolating polynomials. Both of these functions are defined on a grid superposed on the birth-lifetime plane. We then combine the resulting features with machine learning algorithms to perform supervised classification and regression on several example data sets, including manifold data, shape data , and an embedded time series from a Rossler system. Our results show that the template function approach yields high accuracy rates that match and often exceed the results of existing methods for featurizing persistence diagrams . One counter-intuitive observation is that in most cases using interpolating polynomials, where each point contributes globally to the feature vector, yields significantly better results than using tent functions, where the contribution of each point is localized to its grid cell . Along the way, we also provide a complete characterization of compact sets in persistence diagram space endowed with the bottleneck distance .
The persistence diagram is an increasingly useful tool from Topological Data Analysis , but its use alongside typical machine learning techniques requires mathematical finesse. The most success to date has come from methods that map persistence diagrams into \mathbb{R in a way which maximizes the structure preserved. This process is commonly referred to as featurization. In this paper, we describe a mathematical framework for featurization using template functions. These functions are general as they are only required to be continuous and compactly supported . We discuss two realizations : tent functions , which emphasize the local contributions of points in a persistence diagram, and interpolating polynomials, which capture global pairwise interactions. We combine the resulting features with classification and regression algorithms on several examples including shape data and the Rossler system. Our results show that using template functions yields high accuracy rates that match and often exceed those of existing featurization methods . One counter-intuitive observation is that in most cases using interpolating polynomials, where each point contributes globally to the feature vector, yields significantly better results than using tent functions, where the contribution of each point is localized . Along the way, we provide a complete characterization of compactness in the space of persistence diagrams .
R
arising from the field of
from
R
. However, using these diagrams in conjunction with
, but its use alongside typical
D
some
R
finding methods for turning
methods that map
R
vectors in \mathbb{R
\mathbb{R
R
preserves as much of the space of persistence diagrams as possible,
maximizes the structure preserved. This process is
R
featurizing the persistence diagram space
featurization
R
, have a compact support, and separate points
and compactly supported
R
example realizationsof these functions
realizations
R
and Chybeyshev interpolating polynomials. Both of these functions are defined on a grid superposed on the birth-lifetime plane. We then
, which emphasize the local contributions of points in a persistence diagram, and interpolating polynomials, which capture global pairwise interactions. We
D
machine learning algorithms to perform supervised
R
on several example data sets, including manifold data, shape data , and an embedded time series from a
algorithms on several examples including shape data and the
R
the template function approach
using template functions
R
the results of existing methods for featurizing persistence diagrams
those of existing featurization methods
D
to its grid cell
D
also
R
compact sets in persistence diagram space endowed with the bottleneck distance
compactness in the space of persistence diagrams