A structural model on a hypercube represented by optimal transport

We propose a flexible statistical model for high-dimensional quantitative data on a hypercube. Our model, the structural gradient model (SGM), is based on a one-to-one map on the hypercube that is a solution to an optimal transport problem. As we show with many examples, SGM can describe various dependence structures including correlation and heteroscedasticity. The likelihood function is explicitly expressed without any normalizing constant. Simulation of SGM is achieved through a direct extension of the inverse function method. The maximum likelihood estimation of SGM is reduced to the determinant-maximization known as a convex optimization problem. In particular, a lasso-type estimation is available by adding constraints. SGM is compared with graphical Gaussian models and mixture models.