An approximation of the CP-rank of a partially sampled tensor

We exploit the recent algebraic geometry analyses that study the fundamental conditions on the locations of the sampled entries for finite completability of low-rank sampled tensor to treat the problem of CP-rank approximation for a partially sampled tensor. Particularly, the goal is to approximate the unknown rank based on the locations of the sampled entries, i.e., the sampling pattern, and the rank of an arbitrary given completion. First we provide an upper bound on the unknown CP-rank with probability one assuming that the sampling pattern satisfies the proposed combinatorial properties. However, the proposed combinatorial properties may be hard to verify. Hence, we also provide probabilistic versions of such bounds that hold with high probability assuming that the sampling probability is above a threshold, i.e., we provide the sampling probability that results the sampling pattern satisfies the proposed combinatorial properties with high probability. In addition, these upper bounds can be exactly equal to the unknown CP-rank given the lowest-rank completion. To illustrate how tight our proposed upper bounds are, we have provided some numerical results for the case of two-way tensor, i.e., matrix, in which we applied the nuclear norm minimization to find a low-rank completion of the sampled data and observe that the proposed upper bound is almost equal to the true unknown rank.