  Seeking informative projecting directions has been an important task in
utilizing sliced Wasserstein distance in applications. However, finding these
directions usually requires an iterative optimization procedure over the space
of projecting directions, which is computationally expensive. Moreover, the
computational issue is even more severe in deep learning applications, where
computing the distance between two mini-batch probability measures is repeated
several times. This nested loop has been one of the main challenges that
prevent the usage of sliced Wasserstein distances based on good projections in
practice. To address this challenge, we propose to utilize the
learning-to-optimize technique or amortized optimization to predict the
informative direction of any given two mini-batch probability measures. To the
best of our knowledge, this is the first work that bridges amortized
optimization and sliced Wasserstein generative models. In particular, we derive
linear amortized models, generalized linear amortized models, and non-linear
amortized models which are corresponding to three types of novel mini-batch
losses, named amortized sliced Wasserstein. We demonstrate the favorable
performance of the proposed sliced losses in deep generative modeling on
standard benchmark datasets.
