GENERATIVE LEARNING WITH EULER PARTICLE TRANSPORT

Abstract

We propose an Euler particle transport (EPT) approach for generative learning. The proposed approach is motivated by the problem of finding the optimal transport map from a reference distribution to a target distribution characterized by the Monge-Ampere equation. Interpreting the infinitesimal linearization of the Monge-Ampere equation from the perspective of gradient flows in measure spaces leads to a stochastic McKean-Vlasov equation. We use the forward Euler method to solve this equation. The resulting forward Euler map pushes forward a reference distribution to the target. This map is the composition of a sequence of simple residual maps, which are computationally stable and easy to train. The key task in training is the estimation of the density ratios or differences that determine the residual maps. We estimate the density ratios (differences) based on the Bregman divergence with a gradient penalty using deep density-ratio (difference) fitting. We show that the proposed density-ratio (difference) estimators do not suffer from the "curse of dimensionality" if data is supported on a lower-dimensional manifold. Numerical experiments with multi-mode synthetic datasets and comparisons with the existing methods on real benchmark datasets support our theoretical results and demonstrate the effectiveness of the proposed method.

1. INTRODUCTION

The ability to efficiently sample from complex distributions plays a key role in a variety of prediction and inference tasks in machine learning and statistics (Salakhutdinov, 2015) . The long-standing methodology for learning an underlying distribution relies on an explicit statistical data model, which can be difficult to specify in many applications such as image analysis, computer vision and natural language processing. In contrast, implicit generative models do not assume a specific form of the data distribution, but rather learn a nonlinear map to transform a reference distribution to the target distribution. This modeling approach has been shown to achieve impressive performance in many machine learning tasks (Reed et al., 2016; Zhu et al., 2017) . Generative adversarial networks (GAN) (Goodfellow et al., 2014) In this paper, we propose an Euler particle transport (EPT) approach for learning a generative model by integrating ideas from optimal transport, numerical ODE, density-ratio estimation and deep neural networks. We formulate the problem of generative learning as that of finding a nonlinear transform that pushes forward a reference to the target based on the quadratic Wasserstein distance. Since it is challenging to solve the resulting Monge-Ampère equation, we consider the continuity equation derived from the linearization of the Monge-Ampère equation, which is a gradient flows converging to the target distribution. We solve the Mckean-Vlasov equation associated with the gradient flow using the forward Euler method. The resulting EPT that pushes forward the reference distribution to the target distribution is a composition of a sequence of simple residual maps, which are computationally stable and easy to train. The residual maps are completely determined by the density ratios between the distributions at the current iterations and the target distribution. We estimate density ratios based on the Bregman divergence with a gradient regularizer using deep density-ratio fitting. We establish bounds on the approximation errors due to linearization of the Monge-Ampère equation, Euler discretization of the Mckean-Vlasov equation, and deep density-ratio estimation. Our result on



, variational auto-encoders (VAE) (Kingma & Welling, 2014) and flow-based methods (Rezende & Mohamed, 2015) are important representatives of implicit generative models.

