### 计算机代写|机器学习代写machine learning代考|COMP30027

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|机器学习代写machine learning代考|Kernel Matrices of Large-Dimensional Data

Another less-known but equally important example of the curse of dimensionality in machine learning involves the loss of relevance of (the notion of) Euclidean distance between large-dimensional data vectors. To be more precise, we will see in the sequel that, in an asymptotically nontrivial classification setting (i.e., ensuring that asymptotic classification is neither trivially easy nor impossible), large and numerous data vectors $\mathbf{x}1, \ldots, \mathbf{x}_n \in \mathbb{R}^p$ extracted from a few-class (say two-class) mixture model tend to be asymptotically at equal (Euclidean) distance from one another, irrespective of their corresponding class. Roughly speaking, in this nontrivial setting and under some reasonable statistical assumptions on the $x_i \mathrm{~s}$, we have $$\max {1 \leq i \neq j \leq n}\left{\frac{1}{p}\left|\mathbf{x}_i-\mathbf{x}_j\right|^2-\tau\right} \rightarrow 0$$
for some constant $\tau>0$ as $n, p \rightarrow \infty$, independently of the classes (same or different) of $\mathbf{x}_i$ and $\mathbf{x}_j$ (here the normalization by $p$ is used for compliance with the notations in the remainder of this book and has no particular importance).

This asymptotic behavior is extremely counterintuitive and conveys the idea that classification by standard methods ought not to be doable in this large-dimensional regime. Indeed, in the conventional small-dimensional intuition that forged many of the leading machine learning algorithms of everyday use (such as spectral clustering [Ng et al., 2002, Luxburg, 2007]), two data points are assigned to the same class if they are “close” in Euclidean distance. Here we claim that, when $p$ is large, data pairs are neither close nor far from each other, regardless of their belonging to the same class or not. Despite this troubling loss of individual discriminative power between data pairs, we subsequently show that, thanks to a collective behavior of all data belonging to the same (few and thus large) classes, data classification or clustering is still achievable. Better, we shall see that, while many conventional methods devised from small-dimensional intuitions do fail in this large-dimensional regime, some popular approaches, such as the $\mathrm{Ng}$-Jordan-Weiss spectral clustering method [Ng et al., 2002] or the PageRank semisupervised learning approach [Avrachenkov et al., 2012], still function. But the core reasons for their functioning are strikingly different from the reasons of their initial designs, and they often operate far from optimally.

## 计算机代写|机器学习代写machine learning代考|The Nontrivial Classification Regime

To get a clear picture of the source of Equation (1.3), we first need to clarify what we refer to as the “asymptotically nontrivial” classification setting. Consider the simplest scenario of a binary Gaussian mixture classification: Given a training set $\mathbf{x}_1, \ldots, \mathbf{x}_n \in \mathbb{R}^p$ of $n$ samples independently drawn from the two-class $\left(\mathcal{C}_1\right.$ and $\left.\mathcal{C}_2\right)$ Gaussian mixture,
$$\mathcal{C}_1: \mathbf{x} \sim \mathcal{N}\left(\boldsymbol{\mu}, \mathbf{I}_p\right), \quad \mathcal{C}_2: \mathbf{x} \sim \mathcal{N}\left(-\boldsymbol{\mu}, \mathbf{I}_p+\mathbf{E}\right),$$
each drawn with probability $1 / 2$, for some deterministic $\mu \in \mathbb{R}^p$ and symmetric $\mathbf{E} \in \mathbb{R}^{p \times p}$, both possibly depending on $p$. In the ideal case where $\mu$ and $\mathbf{E}$ are perfectly known, one can devise a (decision optimal) Neyman-Pearson test. For an unknown $\mathbf{x}$, genuinely belonging to $\mathcal{C}_1$, the Neyman-Pearson test to decide on the class of $\mathbf{x}$ reads
Writing $\mathbf{x}=\boldsymbol{\mu}+\mathbf{z}$ for $\mathbf{z} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$, the above test is equivalent to
\begin{aligned} T(\mathbf{x}) \equiv & 4 \boldsymbol{\mu}^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1} \boldsymbol{\mu}+4 \boldsymbol{\mu}^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1} \mathbf{z}+\mathbf{z}^{\top}\left(\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}-\mathbf{I}_p\right) \mathbf{z} \ & +\log \operatorname{det}\left(\mathbf{I}_p+\mathbf{E}\right) \underset{\mathcal{C}_2}{\mathcal{C}_1} \underset{\gtrless}{ } . \end{aligned}
Since $\mathbf{U z}$ for $\mathbf{U} \in \mathbb{R}^{p \times p}$, an eigenvector basis of $\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}$ (and thus of $\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}-$ $\mathbf{I}_p$ ), follows the same distribution as $\mathbf{z}$, the random variable $T(\mathbf{x})$ can be written as the sum of $p$ independent random variables. Further assuming that $|\boldsymbol{\mu}|=O(1)$ with respect to $p$, by Lyapunov’s central limit theorem (e.g., [Billingsley, 2012, Theorem 27.3]) and the fact that $\operatorname{Var}\left[\mathbf{z}^{\top} \mathbf{A z}\right]=2 \operatorname{tr}\left(\mathbf{A}^2\right)$ for symmetric $\mathbf{A} \in \mathbb{R}^{p \times p}$ and Gaussian $\mathbf{z}$, we have, as $p \rightarrow \infty$,
$$V_T^{-1 / 2}(T(\mathbf{x})-\bar{T}) \stackrel{d}{\rightarrow} \mathcal{N}(0,1),$$
where
\begin{aligned} \bar{T} & \equiv 4 \mu^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1} \boldsymbol{\mu}+\operatorname{tr}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}-p+\log \operatorname{det}\left(\mathbf{I}_p+\mathbf{E}\right), \ V_T & \equiv 16 \boldsymbol{\mu}^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-2} \boldsymbol{\mu}+2 \operatorname{tr}\left(\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}-\mathbf{I}_p\right)^2 . \end{aligned}

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|The Nontrivial Classification Regime

$$\mathcal{C}_1: \mathbf{x} \sim \mathcal{N}\left(\boldsymbol{\mu}, \mathbf{I}_p\right), \quad \mathcal{C}_2: \mathbf{x} \sim \mathcal{N}\left(-\boldsymbol{\mu}, \mathbf{I}_p+\mathbf{E}\right),$$

$$T(\mathbf{x}) \equiv 4 \boldsymbol{\mu}^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1} \boldsymbol{\mu}+4 \boldsymbol{\mu}^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1} \mathbf{z}+\mathbf{z}^{\top}\left(\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}-\mathbf{I}_p\right) \mathbf{z} \quad+\log \operatorname{det}\left(\mathbf{I}_p+\mathbf{E}\right)$$

[Billingsley, 2012, Theorem 27.3]) 和事实 $\operatorname{Var}\left[\mathbf{z}^{\top} \mathbf{A z}\right]=2 \operatorname{tr}\left(\mathbf{A}^2\right)$ 对于对称 $\mathbf{A} \in \mathbb{R}^{p \times p}$ 和高斯 $\mathbf{z}$, 我 们有 $p \rightarrow \infty$ ，
$$V_T^{-1 / 2}(T(\mathbf{x})-\bar{T}) \stackrel{d}{\rightarrow} \mathcal{N}(0,1),$$

$$\bar{T} \equiv 4 \mu^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1} \boldsymbol{\mu}+\operatorname{tr}\left(\mathbf{I}_p+\mathbf{E}\right)^{-1}-p+\log \operatorname{det}\left(\mathbf{I}_p+\mathbf{E}\right), V_T \quad \equiv 16 \boldsymbol{\mu}^{\top}\left(\mathbf{I}_p+\mathbf{E}\right)^{-2} \boldsymbol{\mu}+$$

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## MATLAB代写

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