Amortized Projection Optimization for Sliced Wasserstein Generative Models

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 minibatch losses, named amortized sliced Wasserstein. We demonstrate the favorable performance of the proposed sliced losses in deep generative modeling on standard benchmark datasets 1 . Recently, optimal transport-based losses, which we denote as D(•, •), are utilized to train generative models due to their training stability, efficiency, and geometrically meaning. Examples of these models include Wasserstein GAN [3] with the dual form of Wasserstein-1 distance [46] , and 51] with the primal form of Wasserstein distance and with Sinkhorn divergence [8]