We discuss structured Schatten norms for tensor decomposition that includes two recently proposed norms ("overlapped" and "latent") for convex-optimization-based tensor decomposition, and connect tensor decomposition with wider literature on structured sparsity. Based on the properties of the structured Schatten norms, we mathematically analyze the performance of "latent" approach for tensor decomposition, which was empirically found to perform better than the "overlapped" approach in some settings. We show theoretically that this is indeed the case. In particular, when the unknown true tensor is low-rank in a specific mode, this approach performs as good as knowing the mode with the smallest rank. Along the way, we show a novel duality result for structures Schatten norms, establish the consistency, and discuss the identifiability of this approach. We confirm through numerical simulations that our theoretical prediction can precisely predict the scaling behavior of the mean squared error.

本文探讨了用于张量分解的结构化Schatten范数（包括“重叠”和“潜在”两个最近提出的范数），将张量分解与结构稀疏性的更广泛文献联系起来，并从经验上发现'Schatten'的结构分析范数性能较好的“潜在”方法进行数学分析。

基于结构化瑟曲范数正则化的凸张量分解