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

2022年12月27日

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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
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## 计算机代写|机器学习代写machine learning代考|Theory versus Practice

Our first argument follows after numerous comparative experiments made between theoretical findings on Gaussian versus real data. Indeed, although mostly derived under simple and seemingly unrealistic Gaussian mixture models, many theoretical results mentioned above show an unexpected close match when applied to popular real-world (sometimes not so) large-dimensional datasets, such as the MNIST handwritten-digit dataset [LeCun et al., 1998], the related Fashion-MNIST [Xiao et al., 2017], Kannada-MNIST [Prabhu, 2019] and Kuzushiji-MNIST [Clanuwat et al., 2018] datasets, the German Traffic Sign dataset [Houben et al., 2013], deep neural network features of the now popular ImageNet dataset [Deng et al., 2009], used for state-of-the-art machine learning and computer vision applications, as well as numerous financial and electroencephalography (EEG) time series datasets. In particular, while most elementary machine learning methods discussed in this book cannot be applied directly on raw ImageNet images to yield satisfactory performance, when performed on “deep” features of the data (such as VGG, DenseNet, or ResNet features) obtained from independent deep neural networks, these algorithms tend to behave the same as with simple Gaussian mixtures [Seddik et al., 2020]. These seemingly striking empirical observations are indeed theoretically sustained by universality arguments arising from the powerful concentration of measure theory.

To be more precise, the following systematic comparison approach will be pursued in this book. An asymptotically nontrivial classification or regression problem is studied: that is, we assume that the problem at hand is theoretically neither too easy nor too hard to solve (as the one discussed in Section 1.1.3) and practically leads, in general, to, say, (binary) classification error rates of the order of $5 \%-30 \%$ and of relative regression errors also of the order $5 \%-30 \%$. In particular, we insist that the asymptotic random matrix framework under study is, in general, incapable to thinly grasp error rates below the $1 \%-2 \%$ region, which may be the domain of “outliers” and marginal data.

Having posed this nontriviality assumption, we shall generically model the data as being drawn from a simple mixture model, for example, the Gaussian mixture model that gives access to a large panoply of powerful technical tools. The theoretical results obtained from the proposed analyses (asymptotic performance notably) are thus function of the statistical means and covariances of the mixture distribution. To compare the theoretical results to real data, we then conduct the following procedure:
(i) exploiting the numerous and labeled samples of the real datasets (such as the $\sim 60000$ images of the training MNIST database), we empirically estimate the scalar functions of the statistical means and covariances (that determine the asymptotic performance of the method under study), for each class in the database;
(ii) we then evaluate the asymptotic performance that a genuine Gaussian mixture model having these means and covariances would have;
(iii) we compare these “theoretical” values to actual simulations.

## 计算机代写|机器学习代写machine learning代考|Concentrated Random Vectors and Real Data Modeling

The modeling assumption that the data vectors $\mathbf{x}i$ are linear or affine $\operatorname{maps~}{\mathbf{x}_i}=\mathbf{A} \mathbf{z}_i+\mathbf{b}$ of random vectors $\mathbf{z}_i$ constituted of i.i.d. entries is simultaneously an asset for random matrix analysis (by exploiting the degrees of freedom in the entries of $\mathbf{z}_i$ ) but a severe practical limitation, as few real datasets are likely of this simplistic form.

El Karoui [2009] provided a first means for random matrix theory to go beyond the “vector of independent entries” assumption. ${ }^9$ There, relying on elements of the concentration of measure theory, extensively developed by Ledoux [2005], El Karoui essentially shows (in a rather technical manner) that some of the early random matrix results from Pastur, Bai, and Silverstein remain valid under the assumption that the $\mathbf{x}_i \mathrm{~s}$ are concentrated random vectors. Roughly speaking, a random vector $\mathbf{x} \in \mathbb{R}^p$ is concentrated if, for a certain family of functions $f: \mathbb{R}^p \rightarrow \mathbb{R}$, there exists a deterministic scalar $M_f \in \mathbb{R}$ such that
$$\mathbb{P}\left(\left|f(\mathbf{x})-M_f\right|>t\right) \leq \alpha(t)$$
for some decreasing function $\alpha: \mathbb{R} \rightarrow \mathbb{R}$; in general, $\alpha(t)$ will be of the form $\alpha(t)=C e^{-c t^q}$ for some $q>0$ and $C, c>0$ constants (which may depend on $p$ though). Intuitively, a concentrated random vector is a (random) point in high-dimensional space having “predictable scalar observation” $f(\mathbf{x})$, in the sense that, with (exponentially) high probability, $f(\mathbf{x})$ takes values very close to the deterministic $M_f$. Thus, in the (one-dimensional) “observable world,” the observation $f(\mathbf{x})$, which may typically be any performance metric of a machine learning algorithm on a test datum $\mathbf{x}$, appears to be “stable” for any concentrated vector $\mathbf{x} .{ }^{10}$

Ledoux and El Karoui mostly focused on concentrated random vectors defined on Lipschitz classes of functions $f$, that is, $\mathbf{x}$ is Lipschitz-concentrated if (1.14) holds for all $f$ such that $|f(\mathbf{x})-f(\mathbf{y})| \leq|\mathbf{x}-\mathbf{y}|$ for all $\mathbf{x}, \mathbf{y} \in \mathbb{R}^p$. These stringent constraints, however, make it hard to find random vector belonging to this class. As a matter of fact, in this class, the only standard random vectors are the Gaussian random vector $\mathbf{x} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$ and the uniform vector on the sphere $\mathbf{u}=\mathbf{x} /|\mathbf{x}| \sim \mathbb{S}^{p-1}$ for $\mathbf{x} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$. However, quite importantly, every $\mathbb{R}^p \rightarrow \mathbb{R}^q$ Lipschitz-mapping $g(\mathbf{x})$ and $g(\mathbf{u})$ of these two random vectors, by definition, also belong to the class. ${ }^{11}$

A visual representation of the notion of concentration is presented in Figure 1.6. Yet, since the widest class of (Lipschitz) concentrated random vectors is restricted to Lipschitz maps of standard Gaussian vectors, at first sight, concentrated random vectors are seemingly no more elaborate models than linear and affine maps of Gaussian vectors. As a consequence, there is a priori no reason to assume that the mixtures of concentrated random vectors can model real data any better than Gaussian mixtures.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Theory versus Practice

（i）利用真实数据集的大量标记样本（例如∼60000训练 MNIST 数据库的图像），我们根据经验估计数据库中每个类的统计均值和协方差（确定所研究方法的渐近性能）的标量函数；
(ii) 然后我们评估具有这些均值和协方差的真正高斯混合模型的渐近性能；
(iii) 我们将这些“理论”值与实际模拟进行比较。

## 计算机代写|机器学习代写machine learning代考|Concentrated Random Vectors and Real Data Modeling

$\operatorname{maps} \mathbf{x}_i=\mathbf{A} \mathbf{z}_i+\mathbf{b}$ 随机向量 $\mathbf{z}_i$ 由 $\mathrm{iid}$ 条目构成同时 是随机矩阵分析的资产 (通过利用条目中的自由度 $\mathbf{z}_i$ ) 但 这是一个严重的实际限制，因为很少有真实数据集可能 是这种简单形式。

El Karoui [2009] 为随机矩阵理论提供了超越”独立条目 向量”假设的第一种方法。 ${ }^9$ 在那里，依靠 Ledoux [2005] 广泛发展的测度集中理论的要素，El Karoui 基本 上表明 (以一种相当技术性的方式) Pastur、Bai 和 Silverstein 的一些早期随机矩阵结果在假设 $\mathbf{x}_i$ s 是集中 的随机向量。粗略地说，一个随机向量 $\mathbf{x} \in \mathbb{R}^p$ 是集中 的，如果，对于某个函数族 $f: \mathbb{R}^p \rightarrow \mathbb{R}$ ，存在一个确定 性标量 $M_f \in \mathbb{R}$ 这样
$$\mathbb{P}\left(\left|f(\mathbf{x})-M_f\right|>t\right) \leq \alpha(t)$$

Ledoux 和 El Karoui 主要关注在 Lipschitz 类函数上定 义的集中随机向量 $f$ ，那是， $\mathbf{x}$ 如果 (1.14) 对所有都成 立，则 Lipschitz 集中 $f$ 这样 $|f(\mathbf{x})-f(\mathbf{y})| \leq|\mathbf{x}-\mathbf{y}|$ 对所有人 $\mathbf{x}, \mathbf{y} \in \mathbb{R}^p$. 然而，这些严格的限制使得很难找 到属于此类的随机向量。事实上，在这个类中，唯一的 标准随机向量是高斯随机向量 $\mathbf{x} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$ 和球体上 的均匀矢量 $\mathbf{u}=\mathbf{x} /|\mathbf{x}| \sim \mathbb{S}^{p-1}$ 为了 $\mathbf{x} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$. 然而，非常重要的是，每 $\mathbb{R}^p \rightarrow \mathbb{R}^q \operatorname{Lipschitz}$ 映射 $g(\mathbf{x})$ 和 $g(\mathbf{u})$ 根据定义，这两个随机向量也属于该类。 11

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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。