In pattern classification, polynomial classifiers are well-studied methods as they are capable of generating complex decision surfaces. Unfortunately, the use of multivariate polynomials is limited to kernels as in support-vector machines, because polynomials quickly become impractical for high-dimensional problems. In this paper, we effectively overcome the curse of dimensionality by employing the tensor train (TT) format to represent a polynomial classifier. Based on the structure of TTs, two learning algorithms are proposed, which involve solving different optimization problems of low computational complexity. Furthermore, we show how both regularization to prevent overfitting and parallelization, which enables the use of large training sets, are incorporated into these methods. The efficiency and efficacy of our tensor-based polynomial classifier are then demonstrated on the two popular data sets U.S. Postal Service and Modified NIST.