Generative Learning of Continuous Data by Tensor Networks
Alex Meiburg Jing Chen Jacob Miller Raphaëlle Tihon Guillaume Rabusseau Alejandro Perdomo-Ortiz
Beyond their origin in modeling many-body quantum systems, tensor networks
have emerged as a promising class of models for solving machine learning
problems, notably in unsupervised generative learning. While possessing many
desirable features arising from their quantum-inspired nature, tensor network
generative models have previously been largely restricted to binary or
categorical data, limiting their utility in real-world modeling problems. We
overcome this by introducing a new family of tensor network generative models
for continuous data, which are capable of learning from distributions
containing continuous random variables. We develop our method in the setting of
matrix product states, first deriving a universal expressivity theorem proving
the ability of this model family to approximate any reasonably smooth
probability density function with arbitrary precision. We then benchmark the
performance of this model on several synthetic and real-world datasets, finding
that the model learns and generalizes well on distributions of continuous and
discrete variables. We develop methods for modeling different data domains, and
introduce a trainable compression layer which is found to increase model
performance given limited memory or computational resources. Overall, our
methods give important theoretical and empirical evidence of the efficacy of
quantum-inspired methods for the rapidly growing field of generative learning.