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Tensor network methods have been a key ingredient of advances in condensedmatter physics and have recently sparked interest in the machine learningcommunity for their ability to compactly represent very high-dimensionalobjects. Tensor network methods can for example be used to efficiently learnlinear models in exponentially large feature spaces [Stoudenmire and Schwab,2016]. In this work, we derive upper and lower bounds on the VC dimension andpseudo-dimension of a large class of tensor network models for classification,regression and completion. Our upper bounds hold for linear modelsparameterized by arbitrary tensor network structures, and we derive lowerbounds for common tensor decomposition models~(CP, Tensor Train, Tensor Ringand Tucker) showing the tightness of our general upper bound. These results areused to derive a generalization bound which can be applied to classificationwith low rank matrices as well as linear classifiers based on any of thecommonly used tensor decomposition models. As a corollary of our results, weobtain a bound on the VC dimension of the matrix product state classifierintroduced in [Stoudenmire and Schwab, 2016] as a function of the so-calledbond dimension~(i.e. tensor train rank), which answers an open problem listedby Cirac, Garre-Rubio and P\'erez-Garc\'ia in [Cirac et al., 2019].

Lower and Upper Bounds on the VC-Dimension of Tensor Network Models