Tensor Train is probably the tensor format most widely used in a number of fields, ranging from physics to machine learning. In this work, we focus on the compression problem of a given multi-way datum. It is known that in Tensor Train the order of the variables to be treated has to be fixed a priori, and this might have a non-negligible influence on the compression performances. We propose a compression method in which a given datum is compressed as a sum of Tensor Trains. Each term of a sum is a Tensor Train in which we do not fix a priori neither the order of variables nor the ranks: these are determined based on error and parsimony criteria, taken into account through the introduction of a trade-off function. We prove that, for finite dimensional data, the trade-off function chosen makes it possible to certify the approximation (in terms of accuracy) and get a precise convergence rate. Several encouraging numerical results were proposed on moderate order tensors (up to 6th order).