In this article, we investigate effective sketching schemes via sparsification for high dimensional multilinear arrays or tensors. More specifically, we propose a novel tensor sparsification algorithm that retains a subset of the entries of a tensor in a judicious way, and prove that it can attain a given level of approximation accuracy in terms of tensor spectral norm with a much smaller sample complexity when compared with existing approaches. In particular, we show that for a kth order d×⋯×d {d}\times \cdots \times {d} cubic tensor of stable rank rs {r}_{ {s}} , the sample size requirement for achieving a relative error ε\varepsilon is, up to a logarithmic factor, of the order r1/2sdk/2/ε {r}_{ {s}}^{1/2} {d}^{ {k}/2} /\varepsilon when ε\varepsilon is relatively large, and rsd/ε2 {r}_{ {s}} {d} /\varepsilon ^{2} and essentially optimal when ε\varepsilon is sufficiently small. It is especially noteworthy that the sample size requirement for achieving a high accuracy is of an order independent of k. To further demonstrate the utility of our techniques, we also study how higher order singular value decomposition (HOSVD) of large tensors can be efficiently approximated via sparsification.

中文翻译：

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通过稀疏化进行有效的张量草图


在本文中，我们通过高维多线性数组或张量的稀疏化研究有效的草图方案。更具体地说，我们提出了一种新颖的张量稀疏算法，该算法以明智的方式保留张量条目的子集，并证明它可以在张量谱范数方面达到给定水平的近似精度，并且样本复杂度要小得多与现有方法相比。特别地，我们表明，对于稳定秩 rs {r}_{ {s}} 的 k 阶 d×⋯×d {d}\times \cdots \times {d} 立方张量，实现相对误差 ε\varepsilon 的量级为对数因子 r1/2sdk/2/ε {r}_{ {s}}^{1/2} {d}^{ {k}/2} /当 ε\varepsilon 相对较大时，rsd/ε2 {r}_{ {s}} {d} /\varepsilon ^{2} 基本上是最优的，当 ε\varepsilon 足够小时。特别值得注意的是，实现高精度所需的样本量要求与 k 无关。为了进一步证明我们技术的实用性，我们还研究了如何通过稀疏化有效地近似大张量的高阶奇异值分解（HOSVD）。