A Class of Topological Pseudodistances for Fast Comparison of
  Persistence Diagrams
Rolando Kindelan Nuñez Mircea Petrache Mauricio Cerda Nancy Hitschfeld
Persistence diagrams (PD)s play a central role in topological data analysis,
and are used in an ever increasing variety of applications. The comparison of
PD data requires computing comparison metrics among large sets of PDs, with
metrics which are accurate, theoretically sound, and fast to compute.
Especially for denser multi-dimensional PDs, such comparison metrics are
lacking. While on the one hand, Wasserstein-type distances have high accuracy
and theoretical guarantees, they incur high computational cost. On the other
hand, distances between vectorizations such as Persistence Statistics (PS)s
have lower computational cost, but lack the accuracy guarantees and in general
they are not guaranteed to distinguish PDs (i.e. the two PS vectors of
different PDs may be equal). In this work we introduce a class of
pseudodistances called Extended Topological Pseudodistances (ETD)s, which have
tunable complexity, and can approximate Sliced and classical Wasserstein
distances at the high-complexity extreme, while being computationally lighter
and close to Persistence Statistics at the lower complexity extreme, and thus
allow users to interpolate between the two metrics. We build theoretical
comparisons to show how to fit our new distances at an intermediate level
between persistence vectorizations and Wasserstein distances. We also
experimentally verify that ETDs outperform PSs in terms of accuracy and
outperform Wasserstein and Sliced Wasserstein distances in terms of
computational complexity.