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：本文引入了一种新的张量范数——平均谱范数，用于研究基于典型多分量分解（CPD）的张量补全问题的样本复杂度。我们探究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实用性。我们的新方法显著降低了基于CPD的含噪张量补全的可证明样本率，为生成对基础均值张量任意精度估计所需的观测含噪条目数提供了迄今最佳上界。在泊松和伯努利多元分布下，我们证明生成含噪观测的$N$阶CPD秩$R$参数张量$\boldsymbol{\mathscr{M}}\in\mathbb{R}^{I\times \cdots\times I}$可通过$\mathcal{O}(IR^2\log^{N+2}(I))$个已揭示条目的大似然估计量逼近。此外，在CPD的非负与正交版本下，我们将结果改进为与秩呈线性依赖关系，达到近最优速率$\mathcal{O}(IR\log^{N+2}(I))$。