  We describe a simple, black-box compression format for tensors with a
multiscale structure. By representing the tensor as a sum of compressed tensors
defined on increasingly coarse grids, we capture low-rank structures on each
grid-scale, and we show how this leads to an increase in compression for a
fixed accuracy. We devise an alternating algorithm to represent a given tensor
in the multiresolution format and prove local convergence guarantees. In two
dimensions, we provide examples that show that this approach can beat the
Eckart-Young theorem, and for dimensions higher than two, we achieve higher
compression than the tensor-train format on six real-world datasets. We also
provide results on the closedness and stability of the tensor format and
discuss how to perform common linear algebra operations on the level of the
compressed tensors.
