Abstract
In this paper, we characterize the deterministic conditions on the locations of the sampled entries, which are equivalent (necessary and sufficient) to finite completability of a tensor given some components of its Tucker rank. In order to derive this characterization, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. Then, using the developed tools for this analysis, we also derive a sufficient condition on the sampling pattern that ensures there exists only one completion for the sampled tensor (unique completability).