High-dimensional data arise naturally in many areas of science and engineering, including machine learning, signal processing, computational physics, and statistics. Such data are often represented as tensors, multi-dimensional generalizations of matrices. While tensors provide a natural representation for multi-modal structure, their direct manipulation quickly becomes challenging as the order grows: the number of parameters increases exponentially, and algebraic expressions involving many indices become difficult to interpret and implement. Tensor networks (TNs) provide an effective framework for addressing these challenges. Originally introduced by Penrose and developed extensively in quantum physics, the graphical language of tensor networks encodes contractions as edges in a graph, reducing notational overhead and revealing structural properties obscured by index notation. Despite the central role of high-dimensional tensors in modern machine learning and numerical analysis, tensor network diagrams remain underutilized outside quantum computing, partly due to the lack of a self-contained mathematical reference accessible to a broad technical audience. This manuscript provides a self-contained guide to tensor networks and their use in tensor algebra. We present the main operations on tensors, contractions, products, and reshaping through, graphical notation, and show how classical tensor decompositions and related computations are naturally expressed in this framework. We also illustrate how tensor networks simplify the derivation of gradients and the manipulation of high-dimensional probability distributions. Throughout, we show that the diagrammatic approach yields genuinely shorter and more transparent proofs of classical identities, rank bounds, and gradient formulas that would otherwise require laborious index manipulation.

翻译：高维数据在科学与工程的众多领域中自然产生，包括机器学习、信号处理、计算物理学和统计学。此类数据通常表示为张量，即矩阵的多维推广。虽然张量为多模态结构提供了自然的表示方式，但随着阶数的增加，其直接操作很快变得具有挑战性：参数数量呈指数增长，涉及多个索引的代数表达式难以解释和实现。张量网络（TN）为应对这些挑战提供了有效的框架。最初由彭罗斯引入并在量子物理学中广泛发展的张量网络图解语言将收缩编码为图中的边，减少了符号开销，并揭示了被索引符号所掩盖的结构特性。尽管高维张量在现代机器学习和数值分析中扮演核心角色，张量网络图在量子计算之外仍未得到充分利用，部分原因是缺乏可供广大技术受众使用的自包含数学参考资料。本手稿提供了关于张量网络及其在张量代数中应用的自包含指南。我们通过图解符号展示了张量的主要操作——收缩、乘积和重塑，并展示了经典张量分解及相关计算如何在该框架中自然表达。我们还阐述了张量网络如何简化梯度推导和高维概率分布的操纵。通篇表明，图解方法能生成比传统方式更简短、更透明的经典恒等式、秩界和梯度公式证明，而传统方式则需要繁琐的索引操作。