We introduce a new tensor norm, the average spectrum norm, to study sample complexity of tensor completion problems based on the canonical polyadic decomposition (CPD). Properties of the average spectrum norm and its dual norm are investigated, demonstrating their utility for low-rank tensor recovery analysis. Our novel approach significantly reduces the provable sample rate for CPD-based noisy tensor completion, providing the best bounds to date on the number of observed noisy entries required to produce an arbitrarily accurate estimate of an underlying mean value tensor. Under Poisson and Bernoulli multivariate distributions, we show that an $N$-way CPD rank-$R$ parametric tensor $\boldsymbol{\mathscr{M}}\in\mathbb{R}^{I\times \cdots\times I}$ generating noisy observations can be approximated by large likelihood estimators from $\mathcal{O}(IR^2\log^{N+2}(I))$ revealed entries. Furthermore, under nonnegative and orthogonal versions of the CPD we improve the result to depend linearly on the rank, achieving the near-optimal rate $\mathcal{O}(IR\log^{N+2}(I))$.

翻译：本文引入一种新的张量范数——平均谱范数，以研究基于规范多元分解（CPD）的张量补全问题的样本复杂度。我们探讨了平均谱范数及其对偶范数的性质，证明了它们在低秩张量恢复分析中的有效性。该新方法显著降低了基于CPD的含噪张量补全的可证明采样率，为观测噪声条目数量提供了迄今最优的界，以生成对基础均值张量的任意精确估计。在泊松和伯努利多变量分布下，我们证明：一个生成噪声观测的$N$阶CPD秩$R$参数张量$\boldsymbol{\mathscr{M}}\in\mathbb{R}^{I\times \cdots\times I}$可通过$\mathcal{O}(IR^2\log^{N+2}(I))$个已揭示条目由大似然估计器近似。此外，在CPD的非负和正交版本下，我们将结果改进为与秩线性相关，达到近优速率$\mathcal{O}(IR\log^{N+2}(I))$。