随机矩阵与统计机器学习专题
Topics in Random Matrices and Statistical Machine Learning
摘要：
本论文由两个独立的部分组成：随机矩阵，这构成了本论文的前三分之一，和机器学习，这构成了其余部分。本文的主要结果有：$(m,n,β)$ -Laguerre矩阵和复合Wishart矩阵的逆矩为有限的充要条件; $k$ -最近邻规则分别在度量σ-有限维空间和度量有限维空间中的普适弱相合性和强相合性。在第一部分中，第一章介绍了$(m,n,β)$ -Laguerre矩阵、Wishart矩阵和复合Wishart矩阵及其联合特征值分布。在第二章中，给出了逆矩有限的一个充分必要条件。在第二部分中，第一章介绍了度量维数和微分性质的各种概念，并对Preiss结果的必要部分进行了证明。第二章介绍了统计机器学习中的数学概念，第三章给出了$k$ -最近邻规则，并证明了Stone定理。在第四章和第五章中，我们给出了我们的主要结果和一些可能的未来发展方向。
英文摘要：
This thesis consists of two independent parts: random matrices, which form the first one-third of this thesis, and machine learning, which constitutes the remaining part. The main results of this thesis are as follows: a necessary and sufficient condition for the inverse moments of $(m,n,β)$-Laguerre matrices and compound Wishart matrices to be finite; the universal weak consistency and the strong consistency of the $k$-nearest neighbor rule in metrically sigma-finite dimensional spaces and metrically finite dimensional spaces respectively. In Part I, the Chapter 1 introduces the $(m,n,β)$-Laguerre matrix, Wishart and compound Wishart matrix and their joint eigenvalue distribution. While in Chapter 2, a necessary and sufficient condition to have finite inverse moments has been derived. In Part II, the Chapter 1 introduces the various notions of metric dimension and differentiation property followed by our proof for the necessary part of Preiss' result. Further, Chapter 2 gives an introduction to the mathematical concepts in statistical machine learning and then the $k$-nearest neighbor rule is presented in Chapter 3 with a proof of Stone's theorem. In chapters 4 and 5, we present our main results and some possible future directions based on it.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
