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Generative adversarial networks (GANs) learn the distribution of observedsamples through a zero-sum game between two machine players, a generator and adiscriminator. While GANs achieve great success in learning the complexdistribution of image, sound, and text data, they perform suboptimally inlearning multi-modal distribution-learning benchmarks including Gaussianmixture models (GMMs). In this paper, we propose Generative AdversarialTraining for Gaussian Mixture Models (GAT-GMM), a minimax GAN framework forlearning GMMs. Motivated by optimal transport theory, we design the zero-sumgame in GAT-GMM using a random linear generator and a softmax-based quadraticdiscriminator architecture, which leads to a non-convex concave minimaxoptimization problem. We show that a Gradient Descent Ascent (GDA) methodconverges to an approximate stationary minimax point of the GAT-GMMoptimization problem. In the benchmark case of a mixture of two symmetric,well-separated Gaussians, we further show this stationary point recovers thetrue parameters of the underlying GMM. We numerically support our theoreticalfindings by performing several experiments, which demonstrate that GAT-GMM canperform as well as the expectation-maximization algorithm in learning mixturesof two Gaussians.

GAT-GMM: Generative Adversarial Training for Gaussian Mixture Models