The limits of this process are optimized partitions, along with their associated optimized partition tree metrics and optimized Haar-like bases for both rows and columns. The optimized tensor-Haar-like bases are constructed through their tensor product. The quality (smoothness) of these optimization stages is measured by the entropy, or sparsity, of the matrix expressed in the current stage tensor-Haar-like bases, which is quantified by the ℓ 1 norm of the tensor-Haar coefficients. It is proved in that the reorganized 2D tensor is bi-Hölder smooth with respect to the joint row-column entries. This point of view permits the definition of gEMD between full 2D -or higher dimensional-tensors, and opens the door to a decomposition of higher dimensional tensors into homogeneous or smooth subblocks. We reiterate that this leads to tensor compression and fast algorithms for numerical multilinear transformations. The details of our implementation specifically for 3D tensors can be found in Appendix B.2.