  Optimal transport (OT) is attracting increasing attention in machine
learning. It aims to transport a source distribution to a target one at minimal
cost. In its vanilla form, the source and target distributions are
predetermined, which contracts to the real-world case involving undetermined
targets. In this paper, we propose Doubly Bounded Optimal Transport (DB-OT),
which assumes that the target distribution is restricted within two boundaries
instead of a fixed one, thus giving more freedom for the transport to find
solutions. Based on the entropic regularization of DB-OT, three scaling-based
algorithms are devised for calculating the optimal solution. We also show that
our DB-OT is helpful for barycenter-based clustering, which can avoid the
excessive concentration of samples in a single cluster. Then we further develop
DB-OT techniques for long-tailed classification which is an emerging and open
problem. We first propose a connection between OT and classification, that is,
in the classification task, training involves optimizing the Inverse OT to
learn the representations, while testing involves optimizing the OT for
predictions. With this OT perspective, we first apply DB-OT to improve the
loss, and the Balanced Softmax is shown as a special case. Then we apply DB-OT
for inference in the testing process. Even with vanilla Softmax trained
features, our extensive experimental results show that our method can achieve
good results with our improved inference scheme in the testing stage.
