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This paper is concerned with the approximation of tensors using tree-basedtensor formats, which are tensor networks whose graphs are dimension partitiontrees. We consider Hilbert tensor spaces of multivariate functions defined on aproduct set equipped with a probability measure. This includes the case ofmultidimensional arrays corresponding to finite product sets. We propose andanalyse an algorithm for the construction of an approximation using only pointevaluations of a multivariate function, or evaluations of some entries of amultidimensional array. The algorithm is a variant of higher-order singularvalue decomposition which constructs a hierarchy of subspaces associated withthe different nodes of the tree and a corresponding hierarchy of interpolationoperators. Optimal subspaces are estimated using empirical principal componentanalysis of interpolations of partial random evaluations of the function. Thealgorithm is able to provide an approximation in any tree-based format witheither a prescribed rank or a prescribed relative error, with a number ofevaluations of the order of the storage complexity of the approximation format.Under some assumptions on the estimation of principal components, we prove thatthe algorithm provides either a quasi-optimal approximation with a given rank,or an approximation satisfying the prescribed relative error, up to constantsdepending on the tree and the properties of interpolation operators. Theanalysis takes into account the discretization errors for the approximation ofinfinite-dimensional tensors. Several numerical examples illustrate the mainresults and the behavior of the algorithm for the approximation ofhigh-dimensional functions using hierarchical Tucker or tensor train tensorformats, and the approximation of univariate functions using tensorization.

Higher-order principal component analysis for the approximation of tensors in tree-based low-rank formats