## 计算机代写|机器学习代写machine learning代考|COMP4702

2022年12月27日

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• Foundations of Data Science 数据科学基础
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## 计算机代写|机器学习代写machine learning代考|Outline and Online Toolbox

The remainder of the book is divided into two parts.
Chapter 2 introduces the basics of random matrix theory needed for machine learning applications in this book. In doing so, we shall first revisit the traditional approach found in math-oriented sources, such as Bai and Silverstein [2010], based on a Stieltjes transform and truncation machinery, Pastur and Shcherbina [2011], based on a Gaussian-method approach, Tao [2012] and Vershynin [2012], based on concentration inequalities and a nonasymptotic random matrix approach, and also say a few words on Mingo and Speicher [2017], which follows a free probability framework and on Anderson et al. [2010], which is more oriented toward a determinantal point process and large deviations direction. Unlike most of these references though (with the possible exception of Pastur and Shcherbina [2011]), our methodology is primarily centered on the statistical analysis of the resolvent (and only secondarily on the Stieltjes transform) of random matrices, which is the chief object of interest to us in most machine learning applications. The particular mathematical toolbox exploited to derive the results is of secondary importance.
In this chapter, we will successively introduce:

• the fundamental notion of the resolvent $\mathbf{Q}(z)=\left(\mathbf{X}-z \mathbf{I}_n\right)^{-1}$ of a (random) matrix $\mathbf{X}$, and its relations to the eigenvalues of $\mathbf{X}$, the limiting spectrum of $\mathbf{X}$, the eigenvectors and eigenspaces associated with some specific eigenvalues, as well as its relations to bilinear and quadratic forms often met in machine learning applications (linear or kernel regression, linear and quadratic discriminant analysis, support vector machines, as well as some simple neural networks);

## 计算机代写|机器学习代写machine learning代考|Spectral Measure and Stieltjes Transform

The first use of the resolvent $\mathbf{Q}{\mathbf{M}}$ is in its relation to the empirical spectral measure $\mu{\mathbf{M}}$ of the matrix $\mathbf{M}$ under study, through the associated Stieltjes transform $m_{\mu_{\mathbf{M}}}$, which we all define next.

Definition 2 (Empirical spectral measure). For a symmetric matrix $\mathbf{M} \in \mathbb{R}^{n \times n}$, the spectral measure or empirical spectral measure or empirical spectral distribution (e.s.d.) $\mu_{\mathbf{M}}$ of $\mathbf{M}$ is defined as the normalized counting measure of the eigenvalues $\lambda_1(\mathbf{M}), \ldots, \lambda_n(\mathbf{M})$ of $\mathbf{M}$
$$\mu_{\mathbf{M}} \equiv \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_i(\mathbf{M})}$$
Since $\int \mu_{\mathbf{M}}(d x)=1$, the spectral measure $\mu_{\mathbf{M}}$ of a matrix $\mathbf{M} \in \mathbb{R}^{n \times n}$ (random or not) is a probability measure. For (probability) measures, we can define their associated Stieltjes transforms as follows.

Definition 3 (Stieltjes transform). For a real probability measure $\mu$ with support $\operatorname{supp}(\mu)$, the Stieltjes transform $m_\mu(z)$ is defined, for all $z \in \mathbb{C} \backslash \operatorname{supp}(\mu)$, as
$$m_\mu(z) \equiv \int \frac{1}{t-z} \mu(d t)$$

This definition and the Stieltjes transform framework in effect extend beyond probability measures to $\sigma$-finite real measures (i.e., measures $\mu$ such that $\mu(\mathbb{R})<\infty$ ), which will occasionally be discussed in this book.

The Stieltjes transform $m_\mu$ has numerous interesting properties: it is complex analytic on its domain of definition $\mathbb{C} \backslash \operatorname{supp}(\mu)$, it is bounded $\left|m_\mu(z)\right| \leq$ $1 / \operatorname{dist}(z, \operatorname{supp}(\mu))$, it satisfies $\mathcal{S}[z]>0 \Rightarrow \mathfrak{S}[m(z)]>0$, and it is an increasing function on all connected components of its restriction to $\mathbb{R} \backslash \operatorname{supp}(\mu)$ (since $m_\mu^{\prime}(x)=$ $\int(t-x)^{-2} \mu(d t)>0$ ) with $\lim {x \rightarrow \pm \infty} m\mu(x)=0$ if $\operatorname{supp}(\mu)$ is bounded.

As a transform, $m_\mu$ admits an inverse formula to recover $\mu$, as per the following result.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Outline and Online Toolbox

• 解决方案的基本概念 $\mathbf{Q}(z)=\left(\mathbf{X}-z \mathbf{I}_n\right)^{-1}-$ 个 (随机) 矩阵 $\mathbf{X}$ ，以及它与特征值的关系 $\mathbf{X}$, 的 极限光谱 $\mathbf{X}$ ，与某些特定特征值相关的特征向量 和特征空间，以及它与机器学习应用中经常遇到 的双线性和二次形式的关系（线性或核回归，线 性和二次判别分析，支持向量机，以及一些简单 的神经网络);

## 计算机代写|机器学习代写machine learning代考|Spectral Measure and Stieltjes Transform

$$\mu_{\mathbf{M}} \equiv \frac{1}{n} \sum_{i=1}^n \delta_{\lambda_i(\mathbf{M})}$$

$$m_\mu(z) \equiv \int \frac{1}{t-z} \mu(d t)$$

$$m_\mu(z) \equiv \int \frac{1}{t-z} \mu(d t)$$

Stieltjes 变换 $m_\mu$ 有许多有趣的特性: 它在其定义域上是 复杂的分析 $\mathbb{C} \backslash \operatorname{supp}(\mu)$ ，它是有界的 $\left|m_\mu(z)\right| \leq$ $1 / \operatorname{dist}(z, \operatorname{supp}(\mu))$, 它满足
$\mathcal{S}[z]>0 \Rightarrow \mathfrak{S}[m(z)]>0$, 它是对所有连接组件的递 增函数 $\mathbb{R} \backslash \operatorname{supp}(\mu)$ (自从 $m_\mu^{\prime}(x)=$ $\left.\int(t-x)^{-2} \mu(d t)>0\right)$ 和 $\lim x \rightarrow \pm \infty m \mu(x)=0$ 如果 $\operatorname{supp}(\mu)$ 是有界的。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。