通过球面覆盖逼近张量范数：弥合原始和对偶之间的差距
Approximating Tensor Norms via Sphere Covering: Bridging the Gap Between Primal and Dual
摘要：
矩阵谱范数和核范数有着广泛的应用。将这些范数推广到高阶张量变得越来越重要，但不幸的是，它们是NP-难计算的，甚至是近似的。尽管这两个范数是相互对偶的，但张量核范数的多项式时间算法所获得的最佳已知逼近界比张量谱范数的逼近界更差。在本文中，我们提出了确定性的算法，最好的张量规范的桥梁这一差距。我们的方法不仅提高了核范数的近似界，而且与现有的张量谱范数近似方法相比，数据无关且易于实现。其主要思想是构造一个单位向量的选择，可以近似地表示单位球，换句话说，一个球冠的集合来覆盖球体。为了这个目的，我们明确地构建了几个集合的球形覆盖与可调参数的近似和基数的不同水平的球冠。这些现成的构造具有独立的意义，因为它们为领域和相关问题的各种决策问题提供了强大的工具。我们相信构造的思想和张量范数的近似应用可以用于解决其他集合上的最优化问题，如二元超立方体。
英文摘要：
The matrix spectral and nuclear norms appear in enormous applications. The generalizations of these norms to higher-order tensors is becoming increasingly important but unfortunately they are NP-hard to compute or even approximate. Although the two norms are dual to each other, the best known approximation bound achieved by polynomial-time algorithms for the tensor nuclear norm is worse than that for the tensor spectral norm. In this paper, we bridge this gap by proposing deterministic algorithms with the best bound for both tensor norms. Our methods not only improve the approximation bound for the nuclear norm, but are also data independent and easily implementable comparing to existing approximation methods for the tensor spectral norm. The main idea is to construct a selection of unit vectors that can approximately represent the unit sphere, in other words, a collection of spherical caps to cover the sphere. For this purpose, we explicitly construct several collections of spherical caps for sphere covering with adjustable parameters for different levels of approximations and cardinalities. These readily available constructions are of independent interest as they provide a powerful tool for various decision making problems on spheres and related problems. We believe the ideas of constructions and the applications to approximate tensor norms can be useful to tackle optimization problems over other sets such as the binary hypercube.
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