  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 parameter 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.
