标题： 迷人张量的几何世界与优化挑战：2 $\times$ 2 $\times$ 2 的几何与优化问题 显示英文标题

标题： The Fascinating World of 2 $\times$ 2 $\times$ 2 Tensors: Its Geometry and Optimization Challenges

摘要： 本文献侧重于几何和代数上的复杂性，以区分张量与矩阵，从而补充线性代数、矩阵分析以及张量分解高级课程中的内容。 通过实值的 2 $\times$ 2 $\times$ 2 张量的例子，我们展示了张量如何违背许多矩阵的已知性质：（1）矩阵的秩由其最小维度限制，但一个 2 $\times$ 2 $\times$ 2 张量的秩可以是 3。（2）矩阵只有一个典型秩，而一个通用的 2 $\times$ 2 $\times$ 2 张量的秩可能是 2 或 3——它有两个典型秩。 (3) 矩阵序列中秩为$r$的任何极限点的秩至多为$r$，但秩为 2 的 2$\times$2$\times$2 张量序列的极限点可能是秩 3（更高的秩）。 (4) 矩阵始终存在最佳秩-$r$近似值，但大小为 2$\times$2$\times$2 的秩 3 张量没有最佳秩-2 近似值。 我们利用代数几何和优化工具统一分析矩阵和张量的情况，推导出这些令人惊讶的事实。 为了帮助学生和教育者理解秩约束集合的几何特性，可以通过我们的交互式可视化工具探索矩阵和张量秩的几何性质。 显示英文摘要

摘要： This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, matrix analysis, and tensor decompositions. Using the case of real-valued 2 $\times$ 2 $\times$ 2 tensors, we show how tensors violate many well-known properties of matrices: (1) The rank of a matrix is bounded by its smallest dimension, but a 2 $\times$ 2 $\times$ 2 tensor can be rank 3. (2) Matrices have a single typical rank, but the rank of a generic 2 $\times$ 2 $\times$ 2 tensor can be 2 or 3 - it has two typical ranks. (3) Any limit point of a sequence of matrices of rank $r$ is at most rank $r$, but a limit point of a sequence of 2 $\times$ 2 $\times$ 2 tensors of rank 2 can be rank 3 (a higher rank). (4) Matrices always have a best rank-$r$ approximation, but no rank-3 tensor of size 2 $\times$ 2 $\times$ 2 has a best rank-2 approximation. We unify the analysis of the matrix and tensor cases using tools from algebraic geometry and optimization, providing derivations of these surprising facts. To build intuition for the geometry of rank-constrained sets, students and educators can explore the geometry of matrix and tensor ranks via our interactive visualization tool.