  The need for efficiently comparing and representing datasets with unknown
alignment spans various fields, from model analysis and comparison in machine
learning to trend discovery in collections of medical datasets. We use manifold
learning to compare the intrinsic geometric structures of different datasets by
comparing their diffusion operators, symmetric positive-definite (SPD) matrices
that relate to approximations of the continuous Laplace-Beltrami operator from
discrete samples. Existing methods typically assume known data alignment and
compare such operators in a pointwise manner. Instead, we exploit the
Riemannian geometry of SPD matrices to compare these operators and define a new
theoretically-motivated distance based on a lower bound of the log-Euclidean
metric. Our framework facilitates comparison of data manifolds expressed in
datasets with different sizes, numbers of features, and measurement modalities.
Our log-Euclidean signature (LES) distance recovers meaningful structural
differences, outperforming competing methods in various application domains.
