用图形化方式讲清张量基础运算，让复杂乘法变直观。
Very Basics of Tensors with Graphical Notations: Unfolding, Calculations, and Decompositions
- 用节点和边表示张量乘法，直观展示多维运算
- 涵盖内积、外积、克罗内克积等常见张量操作
- 适合初学者快速掌握张量符号与图示表达
张量网络图（图形化表示）是一种有效工具，通过节点和边图形化地表示多个张量之间的乘法关系。借助这种图形化方法，复杂的张量乘法可被简洁且直观地描述，有助于理解张量积的本质。事实上，大多数矩阵/张量运算，包括内积、外积、哈达玛积、克罗内克积和Khatri-Rao积，均可用图形化表示。这些矩阵/张量运算构成了信号处理与机器学习中矩阵/张量分解的重要基础模块。本文旨在讲解张量的基本概念及其数学符号与图形化表示方法。许多使用张量的论文省略了这些详细定义与解释，对读者造成理解困难。希望本讲义能为这类读者提供帮助。
原文摘要 · Abstract (English)
Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described simply and intuitively, and it also helps to understand the essence of tensor products. In fact, most of matrix/tensor products including inner product, outer product, Hadamard product, Kronecker product, and Khatri-Rao product can be written in graphical notation. These matrix/tensor operations are essential building blocks for the use of matrix/tensor decompositions in signal processing and machine learning. The purpose of this lecture note is to learn the very basics of tensors and how to represent them in mathematical symbols and graphical notation. Many papers using tensors omit these detailed definitions and explanations, which can be difficult for the reader. I hope this note will be of help to such readers.
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