Title: Fundamental Conditions for Low-CP-Rank Tensor Completion,
Abstract: We consider the problem of low canonical polyadic (CP) rank tensor
completion. A completion is a tensor whose entries agree with the observed
entries and its rank matches the given CP rank. We analyze the manifold
structure corresponding to the tensors with the given rank and define a set of
polynomials based on the sampling pattern and CP decomposition. Then, we show
that finite completability of the sampled tensor is equivalent to having a
certain number of algebraically independent polynomials among the defined
polynomials. Our proposed approach results in characterizing the maximum number
of algebraically independent polynomials in terms of a simple geometric
structure of the sampling pattern, and therefore we obtain the deterministic
necessary and sufficient condition on the sampling pattern for finite
completability of the sampled tensor. Moreover, assuming that the entries of
the tensor are sampled independently with probability $p$ and using the
mentioned deterministic analysis, we propose a combinatorial method to derive a
lower bound on the sampling probability $p$, or equivalently, the number of
sampled entries that guarantees finite completability with high probability. We
also show that the existing result for the matrix completion problem can be
used to obtain a loose lower bound on the sampling probability $p$. In
addition, we obtain deterministic and probabilistic conditions for unique
completability. It is seen that the number of samples required for finite or
unique completability obtained by the proposed analysis on the CP manifold is
orders-of-magnitude lower than that is obtained by the existing analysis on the
Grassmannian manifold.