Unsupervised Generative Modeling Using Matrix Product States

Generative modeling, a typical unsupervised learning that makes use of huge amount of unlabeled data, lies in the heart of rapid development of modern machine learning techniques [1]. Different from discriminative tasks such as pattern recognition, the goal of generative modeling is to model the probability distribution of input data and thus be able to generate new samples according to the distribution. At the research frontier of generative modeling, it was used for finding good data representation and dealing with tasks with missing data. Popular generative machine learning models include the Boltzmann Machines (BM) [2, 3] and their generalizations [4], variational autoencoders (VAE) [5], autoregressive models [6, 7], nonlinear density estimations [8-10], and the generative adversarial networks (GAN) [11]. For generative model design, one tries to balance the representational power and efficiency of learning and sampling. There is a long history of relation between generative modeling and physics, especially statistical physics. Some celebrated models, such as Hopfield model [12], and Boltzmann machine [2, 3], are closely related to the Ising model in statistical physics, and its inverse version which learns couplings in the Ising model based on given training configurations [13, 14]. The task of generative modeling also shares many similarities with quantum physics research in the sense that both of them try to model probability distributions in an enormously large space. In the past decades, tensor network (TN) states and algorithms have been shown to be an incredibly potent tool set for studying many-body quantum physics with its power in expressing quantum states relevant to realistic situations [15, 16].