Title: Rank Determination for Low-Rank Data Completion,
Abstract: Recently, fundamental conditions on the sampling patterns have been obtained
for finite completability of low-rank matrices or tensors given the
corresponding ranks. In this paper, we consider the scenario where the rank is
not given and we aim to approximate the unknown rank based on the location of
sampled entries and some given completion. We consider a number of data models,
including single-view matrix, multi-view matrix, CP tensor, tensor-train tensor
and Tucker tensor. For each of these data models, we provide an upper bound on
the rank when an arbitrary low-rank completion is given. We characterize these
bounds both deterministically, i.e., with probability one given that the
sampling pattern satisfies certain combinatorial properties, and
probabilistically, i.e., with high probability given that the sampling
probability is above some threshold. Moreover, for both single-view matrix and
CP tensor, we are able to show that the obtained upper bound is exactly equal
to the unknown rank if the lowest-rank completion is given. Furthermore, we
provide numerical experiments for the case of single-view matrix, where we use
nuclear norm minimization to find a low-rank completion of the sampled data and
we observe that in most of the cases the proposed upper bound on the rank is
equal to the true rank.