SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 275-298, January 2021.
Tensors are often compressed by expressing them in data-sparse tensor formats, where storage costs in such formats are less than those in the original structure. In this paper, we develop three methodologies that bound the compressibility of a tensor: (1) Algebraic structure, (2) smoothness, and (3) displacement structure. For each methodology, we derive bounds on storage costs in various low rank tensor formats that partially explain the abundance of compressible tensors in applied mathematics. For example, using displacement structure, we show that the solution tensor $\mathcal{X} \in \mathbb{C}^{n \times n \times n}$ of a discretized Poisson equation $-\nabla^2 u =1$ on $[-1,1]^3$ with zero Dirichlet conditions can be approximated to a relative accuracy of $0<\epsilon<1$ in the Frobenius norm by a tensor in tensor-train format with $\mathcal{O}(n (\log n)^2 (\log(1/\epsilon))^2)$ degrees of freedom. The constructive bound also allows us to design a spectral algorithm that solves this equation with $\mathcal{O}(n (\log n)^3 (\log(1/\epsilon))^3)$ complexity.

中文翻译：

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关于张量的可压缩性

SIAM 矩阵分析与应用杂志，第 42 卷，第 1 期，第 275-298 页，2021 年 1 月。
张量通常通过以数据稀疏张量格式表达来压缩，这种格式的存储成本低于原始结构中的存储成本。在本文中，我们开发了三种限制张量可压缩性的方法：（1）代数结构，（2）平滑度，以及（3）位移结构。对于每种方法，我们推导出各种低秩张量格式的存储成本界限，这部分解释了应用数学中可压缩张量的丰富性。例如，使用位移结构，我们证明离散泊松方程 $-\nabla^2 u = 的解张量 $\mathcal{X} \in \mathbb{C}^{n \times n \times n}$ $[-1,1]^3$ 上的 1$ 零狄利克雷条件可以近似为 $0<\epsilon< 1$ 在 Frobenius 范数中由张量训练格式的张量计算，具有 $\mathcal{O}(n (\log n)^2 (\log(1/\epsilon))^2)$ 自由度。建设性界限还允许我们设计一个谱算法来求解具有 $\mathcal{O}(n (\log n)^3 (\log(1/\epsilon))^3)$ 复杂度的方程。