Principal component analysis and optimal weighted least-squares method for training tree tensor networks

One of the most challenging tasks in computational science is the approximation of high-dimensional functions. Most of the time, only a few information on the functions is available, and approximating high-dimensional functions requires exploiting low-dimensional structures of these functions. In this work, the approximation of a function u is built using point evaluations of the function, where these evaluations are selected adaptively. Such problems are encountered when the function represents the output of a black-box computer code, a system or a physical experiment for a given value of a set of input variables. This algorithm relies on an extension of principal components analysis (PCA) to multivariate functions in order to estimate the tensors $v_{\alpha}$. In practice, the PCA is realized on sample-based projections of the function u, using interpolation or least-squares regression. Least-squares regression can provide a stable projection but it usually requires a high number of evaluations of u, which is not affordable when one evaluation is very costly. In [1] the authors proposed an optimal weighted least-squares method, with a choice of weights and samples that garantee an approximation error of the order of the best approximation error using a minimal number of samples. We here present an extension of this methodology for the approximation in tree-based format, where optimal weighted least-squares method is used for the projection onto tensor product spaces. This approach will be compared with a strategy using standard least-squares method or interpolation (as proposed in [2]).