  Motivated by the manifold hypothesis, which states that data with a high
extrinsic dimension may yet have a low intrinsic dimension, we develop refined
statistical bounds for entropic optimal transport that are sensitive to the
intrinsic dimension of the data. Our bounds involve a robust notion of
intrinsic dimension, measured at only a single distance scale depending on the
regularization parameter, and show that it is only the minimum of these
single-scale intrinsic dimensions which governs the rate of convergence. We
call this the Minimum Intrinsic Dimension scaling (MID scaling) phenomenon, and
establish MID scaling with no assumptions on the data distributions so long as
the cost is bounded and Lipschitz, and for various entropic optimal transport
quantities beyond just values, with stronger analogs when one distribution is
supported on a manifold. Our results significantly advance the theoretical
state of the art by showing that MID scaling is a generic phenomenon, and
provide the first rigorous interpretation of the statistical effect of entropic
regularization as a distance scale.
