Prevalent matrix completion theories reply on an assumption that the locations of the missing data are distributed uniformly and randomly (i.e., uniform sampling). Nevertheless, the reason for observations being missing often depends on the unseen observations themselves, and thus the missing data in practice usually occurs in a nonuniform and deterministic fashion rather than randomly. To break through the limits of random sampling, this paper introduces a new hypothesis called isomeric condition, which is provably weaker than the assumption of uniform sampling and arguably holds even when the missing data is placed irregularly. Equipped with this new tool, we prove a series of theorems for missing data recovery and matrix completion. In particular, we prove that the exact solutions that identify the target matrix are included as critical points by the commonly used nonconvex programs. Unlike the existing theories for nonconvex matrix completion, which are built upon the same condition as convex programs, our theory shows that nonconvex programs have the potential to work with a much weaker condition. Comparing to the existing studies on nonuniform sampling, our setup is more general.

We investigate a subclass of exponential family graphical models of which the sufficient statistics are defined by arbitrary additive forms.

The recently established RPCA [4] method provides a convenient way to restore low-rank matrices from grossly corrupted observations. While elegant in theory and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when data are strictly low-rank. This is because RPCA ignores clustering structures of the data which are ubiquitous in applications. As the number of cluster grows, the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades.

Prevalent matrix completion theories reply on an assumption that the locations of the missing data are distributed uniformly and randomly (i.e., uniform sampling). Nevertheless, the reason for observations being missing often depends on the unseen observations themselves, and thus the missing data in practice usually occurs in a nonuniform and deterministic fashion rather than randomly. To break through the limits of random sampling, this paper introduces a new hypothesis called isomeric condition, which is provably weaker than the assumption of uniform sampling and arguably holds even when the missing data is placed irregularly. Equipped with this new tool, we prove a series of theorems for missing data recovery and matrix completion. In particular, we prove that the exact solutions that identify the target matrix are included as critical points by the commonly used nonconvex programs. Unlike the existing theories for nonconvex matrix completion, which are built upon the same condition as convex programs, our theory shows that nonconvex programs have the potential to work with a much weaker condition. Comparing to the existing studies on nonuniform sampling, our setup is more general.