  The ability to represent and compare machine learning models is crucial in
order to quantify subtle model changes, evaluate generative models, and gather
insights on neural network architectures. Existing techniques for comparing
data distributions focus on global data properties such as mean and covariance;
in that sense, they are extrinsic and uni-scale. We develop a first-of-its-kind
intrinsic and multi-scale method for characterizing and comparing data
manifolds, using a lower-bound of the spectral variant of the
Gromov-Wasserstein inter-manifold distance, which compares all data moments. In
a thorough experimental study, we demonstrate that our method effectively
discerns the structure of data manifolds even on unaligned data of different
dimensionalities; moreover, we showcase its efficacy in evaluating the quality
of generative models.
