随机张量的谱范数
Spectral norm of random tensors
摘要：
我们证明了随机$n_1\times n_2\times \cdots \times n_K$张量（或高阶数组）的谱范数在某些次高斯假设下的尺度为$O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$。证明是基于一个覆盖数参数。由于谱范数与张量核范数（秩为1的张量集合中最紧的凸松弛）是对偶的，因此该界意味着凸松弛产生的样本复杂度在维数（和）上是线性的，这比最近提出的其他使用展开的张量秩凸松弛小小于。
英文摘要：
We show that the spectral norm of a random $n_1\times n_2\times \cdots \times n_K$ tensor (or higher-order array) scales as $O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right)$ under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor nuclear norm (the tightest convex relaxation of the set of rank one tensors), the bound implies that the convex relaxation yields sample complexity that is linear in (the sum of) the number of dimensions, which is much smaller than other recently proposed convex relaxations of tensor rank that use unfolding.
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