  Tensor network methods have been a key ingredient of advances in condensed
matter physics and have recently sparked interest in the machine learning
community for their ability to compactly represent very high-dimensional
objects. Tensor network methods can for example be used to efficiently learn
linear models in exponentially large feature spaces [Stoudenmire and Schwab,
2016]. In this work, we derive upper and lower bounds on the VC dimension and
pseudo-dimension of a large class of tensor network models for classification,
regression and completion. Our upper bounds hold for linear models
parameterized by arbitrary tensor network structures, and we derive lower
bounds for common tensor decomposition models~(CP, Tensor Train, Tensor Ring
and Tucker) showing the tightness of our general upper bound. These results are
used to derive a generalization bound which can be applied to classification
with low rank matrices as well as linear classifiers based on any of the
commonly used tensor decomposition models. As a corollary of our results, we
obtain a bound on the VC dimension of the matrix product state classifier
introduced in [Stoudenmire and Schwab, 2016] as a function of the so-called
bond dimension~(i.e. tensor train rank), which answers an open problem listed
by Cirac, Garre-Rubio and P\'erez-Garc\'ia in [Cirac et al., 2019].
