In this paper we propose new techniques to sample arbitrary third-order
tensors, with an objective of speeding up tensor algorithms that have recently
gained popularity in machine learning. Our main contribution is a new way to
select, in a biased random way, only $O(n^{1.5}/\epsilon^2)$ of the possible
$n^3$ elements while still achieving each of the three goals: \\ {\em (a)
tensor sparsification}: for a tensor that has to be formed from arbitrary
samples, compute very few elements to get a good spectral approximation, and
for arbitrary orthogonal tensors {\em (b) tensor completion:} recover an
exactly low-rank tensor from a small number of samples via alternating least
squares, or {\em (c) tensor factorization:} approximating factors of a low-rank
tensor corrupted by noise. \\ Our sampling can be used along with existing
tensor-based algorithms to speed them up, removing the computational bottleneck
in these methods.

本文提出了新技术，以加速机器学习中最近流行的张量算法，并通过一种新的随机方式仅选择可能 n^3 元素的 O (n ^ 1.5/ Epsilon ^ 2) 元素，以达到张量稀疏化，张量完成和张量因子化的三个目的。

张量的新采样技术

A New Sampling Technique for Tensors

Within the field of numerical multilinear algebra, block tensors are
increasingly important. Accordingly, it is appropriate to develop an
infrastructure that supports reasoning about block tensor computation. In this
paper we establish concise notation that is suitable for the analysis and
development of block tensor algorithms, prove several useful block tensor
identities, and make precise the notion of a block tensor unfolding.

本文旨在建立适用于块张量算法分析和开发的简洁符号，并证明了几个有用的块张量恒等式，并明确了块张量展开的概念。