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计算机代写|机器学习代写Machine Learning代考|COMP7703 Kernel Methods

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计算机代写|机器学习代写Machine Learning代考|Kernel Methods

Consider a ML (classification or regression) problem with an underlying feature space $\mathcal{X}$. In order to predict the label $y \in \mathcal{Y}$ of a datapoint based on its features $\mathbf{x} \in \mathcal{X}$, we apply a predictor $h$ selected out of some hypothesis space $\mathcal{H}$. Let us assume that the available computational infrastructure only allows us to use a linear hypothesis space $\mathcal{H}^{(n)}$ (see (3.1)).

For some applications, using a linear hypothesis $h(\mathbf{x})=\mathbf{w}^T \mathbf{x}$ is not suitable since the relation between features $\mathbf{x}$ and label $y$ might be highly non-linear. One approach to extend the capabilities of linear hypotheses is to transform the raw features of a data point before applying a linear hypothesis $h$.

The family of kernel methods is based on transforming the features $\mathbf{x}$ to new features $\hat{\mathbf{x}} \in \mathcal{X}^{\prime}$ which belong to a (typically very) high-dimensional space $\mathcal{X}^{\prime}$ [5]. It is not uncommon that, while the original feature space is a low-dimensional Euclidean space (e.g., $\mathcal{X}=\mathbb{R}^2$ ), the transformed feature space $\mathcal{X}^{\prime}$ is an infinite-dimensional function space.

The rationale behind transforming the original features into a new (higherdimensional) feature space $\mathcal{X}^{\prime}$ is to reshape the intrinsic geometry of the feature vectors $\mathbf{x}^{(i)} \in \mathcal{X}$ such that the transformed feature vectors $\hat{\mathbf{x}}^{(i)}$ have a “simpler” geometry (see Fig. 3.7).

计算机代写|机器学习代写Machine Learning代考|Decision Trees

A decision tree is a flowchart-like description of a map $h: \mathcal{X} \rightarrow \mathcal{Y}$ which maps the features $\mathbf{x} \in \mathcal{X}$ of a datapoint to a predicted label $h(\mathbf{x}) \in \mathcal{Y}[6]$. While decision trees can be used for arbitrary feature space $\mathcal{X}$ and label space $\mathcal{Y}$, we will discuss them for the particular feature space $\mathcal{X}=\mathbb{R}^2$ and label space $\mathcal{Y}=\mathbb{R}$.

Figure $3.8$ depicts an example for a decision tree. A decision tree consists of nodes which are connected by directed edges. We can think of a decision tree as a step-by-step instruction, or a “recipe”, for how to compute the function value $h(\mathbf{x})$ given the features $\mathbf{x} \in \mathcal{X}$ of a datapoint. This computation starts at the root node and ends at one of the leaf nodes of the decision tree.

A leaf node $m$, which does not have any outgoing edges, represents a decision region $\mathcal{R}_m \subseteq \mathcal{X}$ in the feature space. The hypothesis $h$ associated with a decision tree is constant over the regions $\mathcal{R}_m$, such that $h(\mathbf{x})=h_m$ for all $\mathbf{x} \in \mathcal{R}_m$ and some fixed number $h_m \in \mathbb{R}$.
In general, there are two types of nodes in a decision tree:

• decision (or test) nodes, which represent particular “tests” about the feature vector $\mathbf{x}$ (e.g., “is the norm of $\mathbf{x}$ larger than 10?”).
• leaf nodes, which correspond to subsets of the feature space.
The particular decision tree depicted in Fig. $3.8$ consists of two decision nodes (including the root node) and three leaf nodes.

5
. 并不少见的是，虽然原始特征空间是低维欧氏空间 $e . g ., \$ \mathcal{X}=\mathbb{R}^2 \$$, 变换后的特征空间 \mathcal{X}^{\prime} 是无限维函数空间。 状seeFig. 3.7. 计算机代写|机器学习代写MACHINE LEARNING代考|DECISION TREES 决策树是对地图的类似流程图的描述 h: \mathcal{X} \rightarrow \mathcal{Y} 映射特征 x \in \mathcal{X} 数据点到预测标签 h(\mathbf{x}) \in \mathcal{Y}[6]. 虽然决策树可用于任意特征空间 \mathcal{X} 和标签空间 \mathcal{Y} ，我们将针对特定 的特征空间讨论它们 \mathcal{X}=\mathbb{R}^2 和标签空间 \mathcal{Y}=\mathbb{R}. 数字 3.8 描述了决策树的示例。决策树由由有向边连接的节点组成。我们可以将决策树视为有关如何计算函数值的分步说明或 “食谱” h(\mathbf{x}) 给定的功能 \mathbf{x} \in \mathcal{X} 的一个 数据点。此计算从根节点开始，到决策树的叶节点之一结束。 叶节点 m ，它没有任何出边，代表一个决策区域 \mathcal{R}_m \subseteq \mathcal{X} 在特征空间中。假设 h 与决策树相关联的区域是恒定的 \mathcal{R}_m, 这样 h(\mathbf{x})=h_m 对所有人 \mathbf{x} \in \mathcal{R}_m 和一些固定 른码 h_m \in \mathbb{R}. 通常，决策树中有两种类型的节点: • 决定ortest节点，代表关于特征向量的特定“测试” \mathbf{x} e.g., “isthenormof \ \mathbf{x} \$$ largerthan $10 ?$ ?”.
• 叶节点，对应于特征空间的子集。
具体的决策树如图 1 所示。 $3.8$ 由两个决策节点组成includingtherootnode和三个叶节点。

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。