Tensor Sandwich: Tensor Completion for Low CP-Rank Tensors via Adaptive Random Sampling

We propose an adaptive and provably accurate tensor completion approach based on combining matrix completion techniques (see, e.g., [1] –[3]) for a small number of slices with a modified noise robust version of Jennrich’s algorithm (see, e.g., [4], [Section 3]). In the simplest case, this leads to a sampling strategy that more densely samples two outer slices (the bread), and then more sparsely samples additional inner slices (the bbq-braised tofu) for the final completion. Under mild assumptions on the factor matrices, the proposed algorithm completes an n × n × n tensor with CP-rank r with high probability while using at most $\mathcal{O}\left( {nr{{\log }^2}r} \right)$ adaptively chosen samples. Empirical experiments further verify that the proposed approach works well in practice, including as a low-rank approximation method in the presence of additive noise.