Point Cloud Classification via Deep Set Linearized Optimal Transport
Scott Mahan Caroline Moosmüller Alexander Cloninger
We introduce Deep Set Linearized Optimal Transport, an algorithm designed for
the efficient simultaneous embedding of point clouds into an $L^2-$space. This
embedding preserves specific low-dimensional structures within the Wasserstein
space while constructing a classifier to distinguish between various classes of
point clouds. Our approach is motivated by the observation that $L^2-$distances
between optimal transport maps for distinct point clouds, originating from a
shared fixed reference distribution, provide an approximation of the
Wasserstein-2 distance between these point clouds, under certain assumptions.
To learn approximations of these transport maps, we employ input convex neural
networks (ICNNs) and establish that, under specific conditions, Euclidean
distances between samples from these ICNNs closely mirror Wasserstein-2
distances between the true distributions. Additionally, we train a
discriminator network that attaches weights these samples and creates a
permutation invariant classifier to differentiate between different classes of
point clouds. We showcase the advantages of our algorithm over the standard
deep set approach through experiments on a flow cytometry dataset with a
limited number of labeled point clouds.