  The tensor data recovery task has thus attracted much research attention in
recent years. Solving such an ill-posed problem generally requires to explore
intrinsic prior structures underlying tensor data, and formulate them as
certain forms of regularization terms for guiding a sound estimate of the
restored tensor. Recent research have made significant progress by adopting two
insightful tensor priors, i.e., global low-rankness (L) and local smoothness
(S) across different tensor modes, which are always encoded as a sum of two
separate regularization terms into the recovery models. However, unlike the
primary theoretical developments on low-rank tensor recovery, these joint L+S
models have no theoretical exact-recovery guarantees yet, making the methods
lack reliability in real practice. To this crucial issue, in this work, we
build a unique regularization term, which essentially encodes both L and S
priors of a tensor simultaneously. Especially, by equipping this single
regularizer into the recovery models, we can rigorously prove the exact
recovery guarantees for two typical tensor recovery tasks, i.e., tensor
completion (TC) and tensor robust principal component analysis (TRPCA). To the
best of our knowledge, this should be the first exact-recovery results among
all related L+S methods for tensor recovery. Significant recovery accuracy
improvements over many other SOTA methods in several TC and TRPCA tasks with
various kinds of visual tensor data are observed in extensive experiments.
Typically, our method achieves a workable performance when the missing rate is
extremely large, e.g., 99.5%, for the color image inpainting task, while all
its peers totally fail in such challenging case.
