Learning with tree tensor networks: Complexity estimates and model selection
چکیده
Tree tensor networks, or tree-based formats, are prominent model classes for the approximation of high-dimensional functions in computational and data science. They correspond to sum-product neural networks with a sparse connectivity associated dimension tree widths given by tuple ranks. The power these models has been proved be (near to) optimal classical smoothness classes. However, an empirical risk minimization framework limited number observations, ranks should selected carefully balance estimation errors. We propose analyze complexity-based selection method we its performance over wide range Given family different trees, ranks, product feature spaces sparsity patterns is (\`a la Barron, Birg\'e, Massart) minimizing penalized risk, penalty depending on complexity class derived from estimates metric entropy networks. This choice yields bound predictor. In least-squares setting, after deriving fast rates convergence show that our strategy minimax adaptive including Sobolev Besov (with isotropic, anisotropic mixed dominating smoothness) analytic functions. discuss role network obtaining several regimes. practice, amplitude calibrated slope heuristics method. Numerical experiments regression setting illustrate strategy.