Learning with tree tensor networks: complexity estimates and model selection

In this paper, we propose and analyze a model selection method for tree tensor networks in an empirical risk minimization framework. Tree tensor networks, or tree-based tensor formats, are prominent model classes for the approximation of high-dimensional functions in numerical analysis and data science. They correspond to sum-product neural networks with a sparse connectivity associated with a dimension partition tree $T$, widths given by a tuple $r$ of tensor ranks, and multilinear activation functions (or units). The approximation power of these model classes has been proved to be near-optimal for classical smoothness classes. However, in an empirical risk minimization framework with a limited number of observations, the dimension tree $T$ and ranks $r$ should be selected carefully to balance estimation and approximation errors. In this paper, we propose a complexity-based model selection strategy \`a la Barron, Birg\'e, Massart. Given a family of model classes, with different trees, ranks and tensor product feature spaces, a model is selected by minimizing a penalized empirical risk, with a penalty depending on the complexity of the model class. After deriving bounds of the metric entropy of tree tensor networks with bounded parameters, we deduce a form of the penalty from bounds on suprema of empirical processes. This choice of penalty yields a risk bound for the predictor associated with the selected model. For classical smoothness spaces, we show that the proposed strategy is minimax optimal in a least-squares setting. In practice, the amplitude of the penalty is calibrated with a slope heuristics method. Numerical experiments in a least-squares regression setting illustrate the performance of the strategy for the approximation of multivariate functions and univariate functions identified with tensors by tensorization (quantization).