  Tensors, which provide a powerful and flexible model for representing
multi-attribute data and multi-way interactions, play an indispensable role in
modern data science across various fields in science and engineering. A
fundamental task is to faithfully recover the tensor from highly incomplete
measurements in a statistically and computationally efficient manner.
Harnessing the low-rank structure of tensors in the Tucker decomposition, this
paper develops a scaled gradient descent (ScaledGD) algorithm to directly
recover the tensor factors with tailored spectral initializations, and shows
that it provably converges at a linear rate independent of the condition number
of the ground truth tensor for two canonical problems -- tensor completion and
tensor regression -- as soon as the sample size is above the order of $n^{3/2}$
ignoring other dependencies, where $n$ is the dimension of the tensor. This
leads to an extremely scalable approach to low-rank tensor estimation compared
with prior art, which suffers from at least one of the following drawbacks:
extreme sensitivity to ill-conditioning, high per-iteration costs in terms of
memory and computation, or poor sample complexity guarantees. To the best of
our knowledge, ScaledGD is the first algorithm that achieves near-optimal
statistical and computational complexities simultaneously for low-rank tensor
completion with the Tucker decomposition. Our algorithm highlights the power of
appropriate preconditioning in accelerating nonconvex statistical estimation,
where the iteration-varying preconditioners promote desirable invariance
properties of the trajectory with respect to the underlying symmetry in
low-rank tensor factorization.
