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In this paper, we analyze the fundamental conditions for low-rank tensorcompletion given the separation or tensor-train (TT) rank, i.e., ranks ofunfoldings. We exploit the algebraic structure of the TT decomposition toobtain the deterministic necessary and sufficient conditions on the locationsof the samples to ensure finite completability. Specifically, we propose analgebraic geometric analysis on the TT manifold that can incorporate the wholerank vector simultaneously in contrast to the existing approach based on theGrassmannian manifold that can only incorporate one rank component. Ourproposed technique characterizes the algebraic independence of a set ofpolynomials defined based on the sampling pattern and the TT decomposition,which is instrumental to obtaining the deterministic condition on the samplingpattern for finite completability. In addition, based on the proposed analysis,assuming that the entries of the tensor are sampled independently withprobability $p$, we derive a lower bound on the sampling probability $p$, orequivalently, the number of sampled entries that ensures finite completabilitywith high probability. Moreover, we also provide the deterministic andprobabilistic conditions for unique completability.

Characterization of Deterministic and Probabilistic Sampling Patterns for Finite Completability of Low Tensor-Train Rank Tensor