Our goal is to develop a more efficient, optimized data compression technique, that can also be extended to higher-dimensional tensor data. For 2D tensors, we begin with an initial metric and initialize the optimization algorithm by constructing two hierarchical partitions (one for the rows and one for the columns). These partitions give rise to two Haar-like orthonormal bases, which in turn lead to updated partition tree metrics. The key idea is that, at each iteration, the partition tree metrics define Hölder smoothness spaces on the rows and columns respectively, with their Hölder norms being equivalent to the ℓ ∞ norm of the Haar coefficients scaled by the folder size. The dual spaces to these Hölder spaces are Besov spaces, characterized by the ℓ 1 norm of the dually-scaled Haar coefficients .