We consider convex relaxations for recovering low-rank tensors based on constrained minimization over a ball induced by the tensor nuclear norm, recently introduced in \cite{tensor_tSVD}. We build on a recent line of results that considered convex relaxations for the recovery of low-rank matrices and established that under a strict complementarity condition (SC), both the convergence rate and per-iteration runtime of standard gradient methods may improve dramatically. We develop the appropriate strict complementarity condition for the tensor nuclear norm ball and obtain the following main results under this condition: 1. When the objective to minimize is of the form $f(\mX)=g(\mA\mX)+\langle{\mC,\mX}\rangle$ , where $g$ is strongly convex and $\mA$ is a linear map (e.g., least squares), a quadratic growth bound holds, which implies linear convergence rates for standard projected gradient methods, despite the fact that $f$ need not be strongly convex. 2. For a smooth objective function, when initialized in certain proximity of an optimal solution which satisfies SC, standard projected gradient methods only require SVD computations (for projecting onto the tensor nuclear norm ball) of rank that matches the tubal rank of the optimal solution. In particular, when the tubal rank is constant, this implies nearly linear (in the size of the tensor) runtime per iteration, as opposed to super linear without further assumptions. 3. For a nonsmooth objective function which admits a popular smooth saddle-point formulation, we derive similar results to the latter for the well known extragradient method. An additional contribution which may be of independent interest, is the rigorous extension of many basic results regarding tensors of arbitrary order, which were previously obtained only for third-order tensors.

翻译：我们考虑基于张量核范数诱导球约束最小化的凸松弛方法，用于恢复低秩张量（该方法近期在文献\tcite{tensor_tSVD}中提出）。基于近期关于低秩矩阵恢复凸松弛方法的研究成果，我们发现：在严格互补条件（SC）下，标准梯度方法的收敛速度和每迭代计算复杂度均可显著提升。本文针对张量核范数球建立恰当的严格互补条件，并在此条件下获得以下主要结果：1. 当最小化目标函数为$f(\mX)=g(\mA\mX)+\langle{\mC,\mX}\rangle$形式时（其中$g$为强凸函数，$\mA$为线性映射，例如最小二乘问题），二次增长界成立，这保证了标准投影梯度方法具有线性收敛速率，尽管$f$本身未必强凸。2. 对于光滑目标函数，若在满足SC条件的最优解附近初始化，标准投影梯度方法仅需计算秩匹配最优解管秩的奇异值分解（用于投影到张量核范数球）。特别地，当管秩为常数时，每迭代计算复杂度近乎与张量规模呈线性关系，而非常规假设下的超线性复杂度。3. 对于具有流行平滑鞍点表达的非光滑目标函数，我们为著名的外梯度方法推导了类似结果。另一个可能具有独立价值的贡献是：我们严格扩展了任意阶张量的多个基本结论，这些结论此前仅针对三阶张量成立。