摘 要：随着机器学习技术的快速发展，线性代数作为其核心数学基础之一，在算法设计与优化中发挥了至关重要的作用。本文旨在系统梳理线性代数在机器学习中的应用现状，并探讨其对算法性能提升的关键贡献。研究通过分析典型机器学习算法的数学模型，揭示矩阵分解、特征值理论及向量空间方法等线性代数工具在数据降维、模型训练和特征提取中的具体实现机制。结果表明，线性代数不仅能够显著提高算法计算效率，还为复杂问题提供了简洁而优雅的数学表达形式。本文创新性地提出了基于线性代数框架的统一算法描述方法，为跨领域算法迁移提供了理论支持。此外，研究进一步总结了当前存在的挑战，如高维数据处理中的计算瓶颈及数值稳定性问题，并对未来发展方向进行了展望，包括结合深度学习探索新型矩阵运算结构的可能性。该研究为机器学习领域的理论发展与实际应用提供了重要参考价值。
关键词：线性代数；机器学习；矩阵分解；特征值理论；算法优化
A Review of Linear Algebra Applications in Machine Learning Algorithms
英文人名
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Abstract：With the rapid development of machine learning technologies, linear algebra, as one of its core mathematical foundations, has played a crucial role in algorithm design and optimization. This paper aims to systematically review the current applications of linear algebra in machine learning and explore its key contributions to enhancing algorithm performance. By analyzing the mathematical models of typical machine learning algorithms, this study elucidates how linear algebra tools such as matrix decomposition, eigenvalue theory, and vector space methods are specifically implemented in data dimensionality reduction, model training, and feature extraction. The results indicate that linear algebra not only significantly improves computational efficiency but also provides concise and elegant mathematical representations for complex problems. Innovatively, this paper proposes a unified algorithm desc ription method based on the fr amework of linear algebra, offering theoretical support for cross-domain algorithm transfer. Furthermore, the study summarizes existing challenges, such as computational bottlenecks and numerical stability issues in high-dimensional data processing, and outlines future directions, including the exploration of novel matrix operation structures in combination with deep learning. This research provides significant reference value for both theoretical advancements and practical applications in the field of machine learning.
Keywords: Linear Algebra；Machine Learning；Matrix Decomposition；Eigenvalue Theory；Algorithm Optimization
目 录
引言 1
一、线性代数基础与机器学习关联 1
(一)向量空间的基本概念 1
(二)矩阵运算的核心作用 2
(三)特征值分解的意义 2
二、数据表示与线性变换的应用 3
(一)数据建模中的向量表示 3
(二)线性变换在降维中的应用 3
(三)主成分分析的数学原理 4
三、优化问题中的线性代数方法 4
(一)梯度下降与矩阵计算 5
(二)最小二乘法的几何解释 5
(三)正定矩阵在优化中的角色 6
四、高级算法中的线性代数技术 6
(一)奇异值分解的应用场景 6
(二)图论中的矩阵表示方法 7
(三)张量分解的初步探索 7
结论 8
参考文献 9
致谢 9
