Tensor factorization has been steadily used in the past decade to represent highdimensional data. In particular, the canonical polyadic (CP) decomposition (CPD) is very appreciated for its modeling power and remarkable uniqueness properties. However, computing the CPD is challenging when the order of the tensor becomes high: numerical issues and high needs for storage and processing can lead the standard algorithms to diverge. To circumvent this limitation, the equivalence between the CPD and the Tensor Train Decomposition (TTD) is exploited. This approach is implemented in a new algorithm called Dimensionality Reduction, joint Estimation of the Ambiguity Matrices and the CP FACtors (DREAMFAC). A global coupled optimization scheme is proposed to break the curse of dimensionality and estimate the CP factors. DREAMFAC performs better than state-of-the-art methods. It avoids the usual propagation of the estimation error in the factors of the TTD. In particular, DREAMFAC reaches the Cramér-Rao lower bounds associated with the considered coupled CP-TT model, which is not the case for the state-of-the-art sequential procedure. Performances are illustrated on the problem of estimating the channels in a dual-polarized MIMO system. Numerical experiments show the competitive performance of the proposed method for recovery of the CP factors and estimation of the channel parameters, even with very low SNR.