1 research outputs found
Topology Applied to Machine Learning: From Global to Local
Through the use of examples, we explain one way in which applied topology has
evolved since the birth of persistent homology in the early 2000s. The first
applications of topology to data emphasized the global shape of a dataset, such
as the three-circle model for pixel patches from natural images,
or the configuration space of the cyclo-octane molecule, which is a sphere with
a Klein bottle attached via two circles of singularity. In these studies of
global shape, short persistent homology bars are disregarded as sampling noise.
More recently, however, persistent homology has been used to address questions
about the local geometry of data. For instance, how can local geometry be
vectorized for use in machine learning problems? Persistent homology and its
vectorization methods, including persistence landscapes and persistence images,
provide popular techniques for incorporating both local geometry and global
topology into machine learning. Our meta-hypothesis is that the short bars are
as important as the long bars for many machine learning tasks. In defense of
this claim, we survey applications of persistent homology to shape recognition,
agent-based modeling, materials science, archaeology, and biology.
Additionally, we survey work connecting persistent homology to geometric
features of spaces, including curvature and fractal dimension, and various
methods that have been used to incorporate persistent homology into machine
learning.Comment: 11 pages, 7 figure