Nuclear Norms for Tensors and Their Use for Convex Multilinear Estimation
Nuclear Norms for Tensors and Their Use for Convex Multilinear Estimation
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发表时间：
2011
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通讯作者：
Marco Signoretto;L. D. Lathauwer;J. Suykens
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作者：
Marco Signoretto;L. D. Lathauwer;J. Suykens
秩约束矩阵问题的一个有前途的凸松弛依赖于核范数的使用，即优化矩阵的奇异值之和。在本文中，我们介绍了一个适当的扩展的概念，Shatten-q范数矩阵-包括核范数-高阶张量。接着，我们研究了一类非光滑凸优化问题，其中张量核范数被用作惩罚函数。我们进一步探讨了多线性秩和新规范之间的链接的优化的后果。从方法论的角度来看，所考虑的一般公式可以专门用于完成不同的（可能是监督的）数据驱动的建模任务。现有的张量完成问题的初步工作可以被视为一个具体的例子，更一般的类的问题在这里考虑。从算法的角度来看，我们开发的主要算法，称为凸多线性估计（CMLE），导致利用分布式凸优化的结果。此外，还提出了这种计算方案的变体，以确保快速收敛，并处理某些等式约束问题。
A promising convex relaxation for rank-constrained matrix problems relies on the use of the nuclear norm, namely the sum of the singular values of the optimization matrix. In this paper we introduce a proper extension of the concept of Shatten-q norms for matrices — including the nuclear norm — to higher order tensors. Successively we study a general class of nonsmooth convex optimization problems where the tensorial nuclear norm is employed as a penalty function. We further explore the consequence for the optimization of the link between multilinear ranks and the new norm. From a methodological perspective, the considered general formulation can be specialized to accomplish different (and possibly supervised) data-driven modeling tasks. Existing preliminary works on the tensor completion problem can be regarded as a specific instance of the more general class of problems considered here. From an algorithmical perspective the main algorithm that we develop, termed Convex MultiLinear Estimation (CMLE), leads to exploiting results from distributed convex optimization. Variants of this computational scheme are additionally proposed to ensure fast convergence and to deal with certain equality constrained problems.