We investigate optimal conditions for inducing low-rankness of higher order tensors by using convex tensor norms with reshaped tensors. We propose the reshaped tensor nuclear norm as a generalized approach to reshape tensors to be regularized by using the tensor nuclear norm. Furthermore, we propose the reshaped latent tensor nuclear norm to combine multiple reshaped tensors using the tensor nuclear norm. We analyze the generalization bounds for tensor completion models regularized by the proposed norms and show that the novel reshaping norms lead to lower Rademacher complexities. Through simulation and real-data experiments, we show that our proposed methods are favorably compared to existing tensor norms consolidating our theoretical claims.

我们研究了通过使用带有重塑张量的凸张量范数来诱导高阶张量的低秩的最佳条件。我们提出了重塑张量核规范，作为重塑张量的通用方法，可以通过使用张量核规范进行正则化。此外，我们提出了重塑的潜在张量核范数，以使用张量核范数组合多个重塑的张量。我们分析了由提出的规范化的张量完成模型的泛化边界，并表明新颖的重塑规范导致较低的Rademacher复杂度。通过仿真和实际数据实验，我们表明，与现有的张量准则巩固了我们的理论主张相比，我们提出的方法具有优势。