Tensor Sandwich: Tensor Completion for Low CP-Rank Tensors via Adaptive Random Sampling
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We propose an adaptive and provably accurate tensor completion approach based on combining matrix completion techniques (see, e.g., arXiv:0805.4471, arXiv:1407.3619, arXiv:1306.2979) for a small number of slices with a modified noise robust version of Jennrich's algorithm. 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 𝒪(nrlog^2 r) 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.
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