Multidimensional signal processing is receiving a lot of interest recently due to the wide spread appearance of multidimensional signals in different applications of data science. Many of these fields rely on prior knowledge of particular properties, such as sparsity for instance, in order to enhance the performance and the efficiency of the estimation algorithms. However, these multidimensional signals are, often, structured into high-order tensors, where the computational complexity and storage requirements become an issue for growing tensor orders. In this paper, we present a sparse-based Joint dImensionality Reduction And Factors rEtrieval (JIRAFE). More specifically, we assume that an arbitrary factor admits a decomposition into a redundant dictionary coded as a sparse matrix, called the sparse coding matrix. The goal is to estimate the sparse coding matrix in the Tensor-Train model framework.