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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## 机器学习代写|流形学习代写manifold data learning代考|Robust Laplacian Eigenmaps Using Global Information

Dimensionality reduction is an important process that is often required to understand the data in a more tractable and humanly comprehensible way. This process has been extensively studied in terms of linear methods such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), Factor Analysis etc. [8]. However, it has been noticed that many high dimensional data, such as a series of related images, lie on a manifold [12] and are not scattered throughout the feature space.

Belkin and Niyogi in [2] proposed Laplacian Eigenmaps (LEM), a method that approximates the Laplace-Beltrami Operator which is able to capture the properties of any Riemaniann manifold. The motivation of our work derives from our experimental observations that when the graph that used Laplacian Eigenmaps (LEM) [2] is not well-constructed (either it has lot of isolated vertices or there are islands of subgraphs) the data is difficult to interpret after a dimension reduction. This paper discusses how global information can be used in addition to local information in the framework of Laplacian Eigenmaps to address such situations. We make use of an interesting result by Costa and Hero that shows that Minimum Spanning Tree on a manifold can reveal its intrinsic dimension and entropy [4]. In other words, it implies that MSTs can capture the underlying global structure of the manifold if it exists. We use this finding to extend the dimension reduction technique using LEM to exploit both local and global information.

LEM depends on the Graph Laplacian matrix and so does our work. Fiedler initially proposed the Graph Laplacian matrix as a means to comprehend the notion of algebraic connectivity of a graph [6]. Merris has extensively discussed the wide variety of properties of the Laplacian matrix of a graph such as invariance, on various bounds and inequalities, extremal examples and constructions, etc., in his survey [10]. A broader role of the Laplacian matrix can be seen in Chung’s book on Spectral Graph Theory [3].

The second section touches on the Graph Laplacian matrix. The role of global information in manifold learning is then presented, followed by our proposed approach of augmenting LEM by including global information about the data. Experimental results confirm that global information can indeed help when the local information is limited for manifold learning.

## 机器学习代写|流形学习代写manifold data learning代考|Definitions

Let us consider a weighted graph $G=(V, E)$, where $V=V(G)=\left{v_{1}, v_{2}, \ldots, v_{n}\right}$ is the set of vertices (also called vertex set) and $E=E(G)=\left{e_{1}, e_{2}, \ldots, e_{n}\right}$ is the set of edges (also called edge set). The weight $w$ function is defined as $w: V \times V \rightarrow \Re$ such that $w\left(v_{i}, v_{j}\right)=w\left(v_{j}, v_{i}\right)=w_{i j}$.

Definition 1: The Laplacian [6] of a graph without loops of multiple edges is defined as the following:
$$L(G)= \begin{cases}d_{v_{i}} & \text { if } v_{i}=v_{j} \ -1 & \text { if } v_{i} \text { are } v_{j} \text { adjacent, } \ 0 & \text { Otherwise. }\end{cases}$$
Fiedler [6] defined the Laplacian of a graph as a symmetric matrix for a regular graph, where $A$ is an adjacency matrix ( $A^{T}$ is the transpose of adjacency matrix), $I$ is the identity matrix, and $n$ is the degree of the regular graph:
$$L(G)=n I-A .$$
A definition by Chung (see [3]) – which is given below – generalizes the Laplacian by adding the weights on the edges of the graph. It can be viewed as Weighed Graph Laplacian. Simply, it is a difference between the diagonal matrix $D$ and $W$, the weighted adjacency matrix.
$$L_{W}(G)=D-W,$$
where the diagonal element in $D$ is defined as $d_{v_{i}}=\sum_{j=1}^{n} w\left(v_{i}, v_{j}\right)$.
Definition 2: The Laplacian of weighted graph (operator) is defined as the following:
$$L_{w}(G)= \begin{cases}d_{v_{i}}-w\left(v_{i}, v_{j}\right) & \text { if } v_{i}=v_{j} \ -w\left(v_{i}, v_{j}\right) & \text { if } v_{i} \text { are } v_{j} \text { connected } \ 0 & \text { otherwise. }\end{cases}$$
$L_{w}(G)$ reduces to $L(G)$ when the edges have unit weights.

# 流形学习代写

## 机器学习代写|流形学习代写manifold data learning代考|Robust Laplacian Eigenmaps Using Global Information

Belkin 和 Niyogi 在 [2] 中提出了 Laplacian Eigenmaps (LEM)，这是一种近似 Laplace-Beltrami 算子的方法，能够捕获任何黎曼流形的属性。我们工作的动机源于我们的实验观察，即当使用拉普拉斯特征图 (LEM) [2] 的图构造不完善（它有很多孤立的顶点或存在子图岛）时，数据很难解释降维后。本文讨论了如何在拉普拉斯特征图框架中使用全局信息和局部信息来解决这种情况。我们利用了 Costa 和 Hero 的一个有趣结果，该结果表明流形上的最小生成树可以揭示其内在维度和熵 [4]。换句话说，这意味着 MST 可以捕获流形的潜在全局结构（如果存在）。我们利用这一发现来扩展使用 LEM 的降维技术，以利用本地和全局信息。

LEM 依赖于图拉普拉斯矩阵，我们的工作也是如此。Fiedler 最初提出图拉普拉斯矩阵作为理解图的代数连通性概念的一种手段 [6]。Merris 在他的调查 [10] 中广泛讨论了图的拉普拉斯矩阵的各种属性，例如不变性、各种边界和不等式、极值示例和构造等。拉普拉斯矩阵的更广泛作用可以在 Chung 的关于谱图理论的书中看到 [3]。

## 机器学习代写|流形学习代写manifold data learning代考|Definitions

\left 的分隔符缺失或无法识别

$$L(G)=\left{\begin{array}{lll} d_{v_{i}} & \text { if } v_{i}=v_{j}-1 \quad \text { if } v_{i} \text { are } v_{j} \text { adjacent, } 0 \quad \text { Otherwise. } \end{array}\right.$$
Fiedler [6] 将图的拉普拉斯算子定义为正则图的对称矩阵，其中 $A$ 是一个邻接矩阵 $\left(A^{T}\right.$ 是邻接矩阵的转置)， $I$ 是单位矩阵，并且 $n$ 是正则图的度数:
$$L(G)=n I-A .$$
Chung 的定义 (见 [3]) 一下面给出一一通过在图的边缘上添加权重来概括拉普拉斯算子。它可以看作是加权图拉普拉斯算子。简单来说，就是对角矩阵的区别 $D$ 和 $W$ ，加权邻接矩阵。
$$L_{W}(G)=D-W,$$

$$L_{w}(G)=\left{d_{v_{i}}-w\left(v_{i}, v_{j}\right) \quad \text { if } v_{i}=v_{j}-w\left(v_{i}, v_{j}\right) \quad \text { if } v_{i} \text { are } v_{j} \text { connected } 0 \quad\right. \text { otherwise. }$$
$L_{w}(G)$ 减少到 $L(G)$ 当边有单位权重时。

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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。

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