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 $n \times n \times n$ tensor with CP-rank $r$ with high probability while using at most $\mathcal{O}(nr\log^2 r)$ 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.

翻译：本文提出一种自适应且具有可证明准确性的张量补全方法，其核心思想是将矩阵补全技术（参见 arXiv:0805.4471、arXiv:1407.3619、arXiv:1306.2979）应用于少量切片，并与改进的 Jennrich 算法（含噪声鲁棒变体）相结合。在最简情形下，该方法采用一种采样策略：对两个外部切片（面包片）进行更密集采样，而对内部切片（烧烤豆腐）进行更稀疏采样，最终完成补全。在关于因子矩阵的温和假设下，所提算法能以高概率完成一个规模为 $n \times n \times n$、CP秩为 $r$ 的张量补全，且仅需使用至多 $\mathcal{O}(nr\log^2 r)$ 个自适应选择的样本。实证实验进一步验证了该方法的实际有效性，包括作为含加性噪声场景下的低秩近似方法。