## 机器学习代写|流形学习代写manifold data learning代考|SCl 7314

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Curves and Geodesics

If the Riemannian manifold $(\mathcal{M}, g)$ is connected, it is a metric space with an induced topology that coincides with the underlying manifold topology. We can, therefore, define a function $d^{\mathcal{M}}$ on $\mathcal{M}$ that calculates distances between points on $\mathcal{M}$ and determines its structure.

Let $\mathbf{p}, \mathbf{q} \in \mathcal{M}$ be any two points on the Riemannian manifold $\mathcal{M}$. We first define the length of a (one-dimensional) curve in $\mathcal{M}$ that joins $\mathbf{p}$ to $\mathbf{q}$, and then the length of the shortest such curve.

A curve in $\mathcal{M}$ is defined as a smooth mapping from an open interval $\Lambda$ (which may have infinite length) in $\Re$ into $\mathcal{M}$. The point $\lambda \in \Lambda$ forms a parametrization of the curve. Let $c(\lambda)=\left(c_{1}(\lambda), \cdots, c_{d}(\lambda)\right)^{\top}$ be a curve in $\Re^{d}$ parametrized by $\lambda \in \Lambda \subseteq \Re$. If we take the coordinate functions, $\left{c_{h}(\lambda)\right}$, of $c(\lambda)$ to be as smooth as needed (usually, $\mathcal{C}^{\infty}$, functions that have any number of continuous derivatives), then we say that $c$ is a smooth curve. If $c(\lambda+\alpha)=c(\lambda)$ for all $\lambda, \lambda+\alpha \in \Lambda$, the curve $c$ is said to be closed. The velocity (or tangent) vector at the point $\lambda$ is given by
$$c^{\prime}(\lambda)=\left(c_{1}^{\prime}(\lambda), \cdots, c_{d}^{\prime}(\lambda)\right)^{\tau},$$
where $c_{j}^{\prime}(\lambda)=d c_{j}(\lambda) / d \lambda$, and the “speed” of the curve is
$$\left|c^{\prime}(\lambda)\right|=\left{\sum_{j=1}^{d}\left[c_{j}^{\prime}(\lambda)\right]^{2}\right}^{1 / 2}$$
Distance on a smooth curve $c$ is given by arc-length, which is measured from a fixed point $\lambda_{0}$ on that curve. Usually, the fixed point is taken to be the origin, $\lambda_{0}=0$, defined to be one of the two endpoints of the data. More generally, the arc-length $L(c)$ along the curve $c(\lambda)$ from point $\lambda_{0}$ to point $\lambda_{1}$ is defined as
$$L(c)=\int_{\lambda_{0}}^{\lambda_{1}}\left|c^{\prime}(\lambda)\right| d \lambda .$$

## 机器学习代写|流形学习代写manifold data learning代考|Linear Manifold Learning

Most statistical theory and applications that deal with the problem of dimensionality reduction are focused on linear dimensionality reduction and, by extension, linear manifold learning. A linear manifold can be visualized as a line, a plane, or a hyperplane, depending upon the number of dimensions involved. Data are observed in some high-dimensional space and it is usually assumed that a lower-dimensional linear manifold would be the most appropriate summary of the relationship between the variables. Although data tend not to live on a linear manifold, we view the problem as having two kinds of motivations. The first such motivation is to assume that the data live close to a linear manifold, the distance off the manifold determined by a random error (or noise) component. A second way of thinking about linear manifold learning is that a linear manifold is really a simple linear approximation to a more complicated type of nonlinear manifold that would probably be a better fit to the data. In both scenarios, the intrinsic dimensionality of the linear manifold is taken to be much smaller than the dimensionality of the data.

Identifying a linear manifold embedded in a higher-dimensional space is closely related to the classical statistics problem of linear dimensionality reduction. The recommended way of accomplishing linear dimensionality reduction is to create a reduced set of linear transformations of the input variables. Linear transformations are projection methods, and so the problem is to derive a sequence of low-dimensional projections of the input data that possess some type of optimal properties.

There are many techniques that can be used for either linear dimensionality reduction or linear manifold learning. In this chapter, we describe only two linear methods, namely, principal component analysis and multidimensional scaling. The earliest projection method was principal component analysis (dating back to 1933), and this technique has become the most popular dimensionality-reducing technique in use today. A related method is that of multidimensional scaling (dating back to 1952), which has a very different motivation. An adaptation of multidimensional scaling provided the core element of the IsOMAP algorithm for nonlinear manifold learning.

## 机器学习代写|流形学习代写manifold data learning代考|Curves and Geodesics

lleft{c_{h}(Nambda)\right }，的 $c(\lambda)$ 尽可能平滑（通常， $\mathcal{C}^{\infty}$ ，具有任意数量的连续导数的函数），那么我们说 $c$ 是 一条平滑曲线。如果 $c(\lambda+\alpha)=c(\lambda)$ 对所有人 $\lambda, \lambda+\alpha \in \Lambda$, 曲线 $c$ 据说是关闭的。该点的速度 (或切线) 矢 量 $\lambda$ 是 (谁) 给的
$$c^{\prime}(\lambda)=\left(c_{1}^{\prime}(\lambda), \cdots, c_{d}^{\prime}(\lambda)\right)^{\tau}$$

$$L(c)=\int_{\lambda_{0}}^{\lambda_{1}}\left|c^{\prime}(\lambda)\right| d \lambda .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|流形学习代写manifold data learning代考|INFS6077

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Topological Spaces

Topological spaces were introduced by Maurice Fréchet (1906) (in the form of metric spaces), and the idea was developed and extended over the next few decades. Amongst those who contributed significantly to the subject was Felix Hausdorff, who in 1914 coined the phrase “topological space” using Johann Benedict Listing’s German word Topologie introduced in $1847 .$

A topological space $\mathcal{X}$ is a nonempty collection of subsets of $\mathcal{X}$ which contains the empty set, the space itself, and arbitrary unions and finite intersections of those sets. A topological space is often denoted by $(\mathcal{X}, \mathcal{T})$, where $\mathcal{T}$ represents the topology associated with $\mathcal{X}$. The elements of $\mathcal{T}$ are called the open sets of $\mathcal{X}$, and a set is closed if its complement is open. Topological spaces can also be characterized through the concept of neighborhood. If $\mathbf{x}$ is a point in a topological space $\mathcal{X}$, its neighborhood is a set that contains an open set that contains $\mathbf{x}$.
Let $\mathcal{X}$ and $\mathcal{Y}$ be two topological spaces, and let $U \subset \mathcal{X}$ and $V \subset \mathcal{Y}$ be open subsets. Consider the family of all cartesian products of the form $U \times V$. The topology formed from these products of open subsets is called the product topology for $\mathcal{X} \times \mathcal{Y}$. If $W \subset \mathcal{X} \times \mathcal{Y}$, then $W$ is open relative to the product topology iff for each point $(x, y) \in \mathcal{X} \times \mathcal{Y}$ there are open neighborhoods, $U$ of $x$ and $V$ of $y$, such that $U \times V \subset W$. For example, the usual topology for $d$-dimensional Euclidean space $\Re^{d}$ consists of all open sets of points in $\Re^{d}$, and this topology is equivalent to the product topology for the product of $d$ copies of $\Re$.

One of the core elements of manifold learning involves the idea of “embedding” one topological space inside another. Loosely speaking, the space $\mathcal{X}$ is said to be embedded in the space $\mathcal{Y}$ if the topological properties of $\mathcal{Y}$ when restricted to $\mathcal{X}$ are identical to the topological properties of $\mathcal{X}$. To be more specific, we state the following definitions. A function $g: \mathcal{X} \rightarrow \mathcal{Y}$ is said to be continuous if the inverse image of an open set in $\mathcal{Y}$ is an open set in $\mathcal{X}$. If $g$ is a bijective (i.e., one-to-one and onto) function such that $g$ and its inverse $g^{-1}$ are continuous, then $g$ is said to be a homeomorphism. Two topological spaces $\mathcal{X}$ and $\mathcal{Y}$ are said to be homeomorphic (or topologically equivalent) if there exists a homeomorphism from one space onto the other. A topological space $\mathcal{X}$ is said to be embedded in a topological space $\mathcal{Y}$ if $\mathcal{X}$ is homeomorphic to a subspace of $\mathcal{Y}$.

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

In the entire theory of topological manifolds, there is no mention of the use of calculus. However, in a prototypical application of a “manifold,” calculus enters in the form of a “smooth” (or differentiable) manifold $\mathcal{M}$, also known as a Riemannian manifold; it is usually defined in differential geometry as a submanifold of some ambient (or surrounding) Euclidean space, where the concepts of length, curvature, and angle are preserved, and where smoothness relates to differentiability. The word manifold (in German, Mannigfaltigkeit) was coined in an “intuitive” way and without any precise definition by Georg Friedrich Bernhard Riemann (1826-1866) in his 1851 doctoral dissertation (Riemann, 1851; Dieudonné, 2009); in 1854, Riemann introduced in his famous Habilitations lecture the idea of a topological manifold on which one could carry out differential and integral calculus.

A topological manifold $\mathcal{M}$ is called a smooth (or differentiable) manifold if $\mathcal{M}$ is continuously differentiable to any order. All smooth manifolds are topological manifolds, but the reverse is not necessarily true. (Note: Authors often differ on the precise definition of a “smooth” manifold.)

We now define the analogue of a homeomorphism for a differentiable manifold. Consider two open sets, $U \in \Re^{r}$ and $V \in \Re^{s}$, and let $g: U \rightarrow V$ so that for $\mathbf{x} \in U$ and $\mathbf{y} \in V, g(\mathbf{x})=$ y. If the function $g$ has finite first-order partial derivatives, $\partial y_{j} / \partial x_{i}$, for all $i=1,2, \ldots, r$, and all $j=1,2, \ldots, s$, then $g$ is said to be a smooth (or differentiable) mapping on $U$. We also say that $g$ is a $\mathcal{C}^{1}$-function on $U$ if all the first-order partial derivatives are continuous. More generally, if $g$ has continuous higher-order partial derivatives, $\partial^{k_{1}+\cdots+k_{r}} y_{j} / \partial x_{1}^{k_{1}} \cdots \partial x_{r}^{k_{r}}$, for all $j=1,2, \ldots, s$ and all nonnegative integers $k_{1}, k_{2}, \ldots, k_{r}$ such that $k_{1}+k_{2}+\cdots+k_{r} \leq r$, then we say that $g$ is a $\mathcal{C}^{\top}$-function, $r=1,2, \ldots$. If $g$ is a $\mathcal{C}^{r}$-function for all $r \geq 1$, then we say that $g$ is a $\mathcal{C}^{\infty}$-function.

If $g$ is a homeomorphism from an open set $U$ to an open set $V$, then it is said to be a $\mathcal{C}^{r}$-diffeomorphism if $g$ and its inverse $g^{-1}$ are both $\mathcal{C}^{r}$-functions. A $\mathcal{C}^{\infty}$-diffeomorphism is simply referred to as a diffeomorphism. We say that $U$ and $V$ are diffeomorphic if there exists a diffeomorphism between them. These definitions extend in a straightforward way to manifolds. For example, if $\mathcal{X}$ and $\mathcal{Y}$ are both smooth manifolds, the function $g: \mathcal{X} \rightarrow \mathcal{Y}$ is a diffeomorphism if it is a homeomorphism from $\mathcal{X}$ to $\mathcal{Y}$ and both $g$ and $g^{-1}$ are smooth. Furthermore, $\mathcal{X}$ and $\mathcal{Y}$ are diffeomorphic if there exists a diffeomorphism between them, in which case, $\mathcal{X}$ and $\mathcal{Y}$ are essentially indistinguishable from each other.

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

Maurice Fréchet (1906) 引入了拓扑空间（以度量空间的形式），这个想法在接下来的几十年中得到发展和扩 展。对这个主题做出重大贡献的人中有 Felix Hausdorff，他在 1914 年使用 Johann Benedict Listing 的德语单词 Topologie 创造了“拓扑空间”一词。1847.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|流形学习代写manifold data learning代考|EECS 559a

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Spectral Embedding Methods for Manifold Learning

Manifold learning encompasses much of the disciplines of geometry, computation, and statistics, and has become an important research topic in data mining and statistical learning. The simplest description of manifold learning is that it is a class of algorithms for recovering a low-dimensional manifold embedded in a high-dimensional ambient space. Major breakthroughs on methods for recovering low-dimensional nonlinear embeddings of highdimensional data (Tenenbaum, de Silva, and Langford, 2000; Roweis and Saul, 2000) led to the construction of a number of other algorithms for carrying out nonlinear manifold learning and its close relative, nonlinear dimensionality reduction. The primary tool of all embedding algorithms is the set of eigenvectors associated with the top few or bottom few eigenvalues of an appropriate random matrix. We refer to these algorithms as spectral embedding methods. Spectral embedding methods are designed to recover linear or nonlinear manifolds, usually in high-dimensional spaces.

Linear methods, which have long been considered part-and-parcel of the statistician’s toolbox, include PRINCIPAL COMPONENT ANALYSIS (PCA) and MULTIDIMENSIONAL SCALING (MDS). PCA has been used successfully in many different disciplines and applications. In computer vision, for example, PCA is used to study abstract notions of shape, appearance, and motion to help solve problems in facial and object recognition, surveillance, person tracking, security, and image compression where data are of high dimensionality (Turk and Pentland, 1991; De la Torre and Black, 2001). In astronomy, where very large digital sky surveys have become the norm, PCA has been used to analyze and classify stellar spectra, carry out morphological and spectral classification of galaxies and quasars, and analyze images of supernova remnants (Steiner, Menezes, Ricci, and Oliveira, 2009). In bioinformatics, PCA has been used to study high-dimensional data generated by genome-wide, gene-expression experiments on a variety of tissue sources, where scatterplots of the top principal components in such studies often show specific classes of genes that are expressed by different clusters of distinctive biological characteristics (Yeung and Ruzzo, 2001; ZhengBradley, Rung, Parkinson, and Brazma, 2010). PCA has also been used to select an optimal subset of single nucleotide polymorphisms (SNPs) (Lin and Altman, 2004). PCA is also used to derive approximations to more complicated nonlinear subspaces, including problems involving data interpolation, compression, denoising, and visualization.

## 机器学习代写|流形学习代写manifold data learning代考|Spaces and Manifolds

Manifold learning involves concepts from general topology and differential geometry. Good introductions to topological spaces include Kelley (1955), Willard (1970), Bourbaki (1989), Mendelson (1990), Steen (1995), James (1999), and several of these have since been reprinted. Books on differential geometry include Spivak (1965), Kreyszig (1991), Kühnel (2000), Lee (2002), and Pressley (2010).

Manifolds generalize the notions of curves and surfaces in two and three dimensions to higher dimensions. Before we give a formal description of a manifold, it will be helpful to visualize the notion of a manifold. Imagine an ant at a picnic, where there are all sorts of items from cups to doughnuts. The ant crawls all over the picnic items, but because of its tiny size, the ant sees everything on a very small scale as flat and featureless. Similarly, a human, looking around at the immediate vicinity, would not see the curvature of the earth. A manifold (also referred to as a topological manifold) can be thought of in similar terms, as a topological space that locally looks flat and featureless and behaves like Euclidean space. Unlike a metric space, a topological space has no concept of distance. In this Section, we review specific definitions and ideas from topology and differential geometry that enable us to provide a useful definition of a manifold.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|流形学习代写manifold data learning代考|Preserving the Estimated Density

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|The Optimization

Now that we have a method to estimate the density on a submanifold of $\mathbb{R}^{D}$, we can proceed to define an algorithm for density preserving maps. ${ }^{9}$ Suppose we are given a sample $X=\left{x_{1}, x_{2}, \ldots, x_{m}\right}$ of $m$ data points $x_{i} \in \mathbb{R}^{D}$ that live on a $d$-dimensional submanifold $M$ of $\mathbb{R}^{D}$. We first proceed to estimate the density at each one of the points, by using a slightly generalized version of the submanifold estimator that has variable bandwidths. Denoting the bandwidth for a given evaluation point $x_{j}$ and a reference (data) point $x_{i}$ by $h_{i j}$, the generalized, variable bandwidth estimator at $x_{j}$ is, ${ }^{10}$
$$\hat{f_{j}}=\hat{f}\left(x_{j}\right)=\frac{1}{m} \sum_{i} \frac{1}{h_{i j}^{d}} K_{d}\left(\frac{\left|x_{j}-x_{i}\right|_{D}}{h_{i j}}\right) .$$
Variable bandwidth methods allow the estimator to adapt to the inhomogeneities in the data. Various approaches exist for picking the bandwidths $h_{i j}$ as functions of the query (evaluation) point $x_{j}$ and/or the reference point $x_{i}[25]$. Here, we focus on the $k$ th-nearest neighbor approach for evaluation points, i.e., we take $h_{i j}$ to depend only on the evaluation point $x_{j}$, and we let $h_{i j}=h_{j}=$ the distance of the $k$ th nearest data (reference) point to the evaluation point $x_{j}$. Here, $k$ is a free parameter that needs to be picked by the user. However, instead of tuning it by hand, one can use a leave-one-out cross-validation score [25] such as the log-likelihood score for the density estimate to pick the best value. This is done by estimating the log-likelihood of each data point by using the leave-one-out version

of the density estimate $(3.7)$ for a range of $k$ values, and picking the $k$ that gives the highest log-likelihood.

Now, given the estimates $\hat{f}{j}=\hat{f}\left(x{j}\right)$ of the submanifold density at the $D$-dimensional data points $x_{j}$, we want to find a $d$-dimensional representation $X^{\prime}=\left{x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{m}^{\prime}\right}$, $x_{i}^{\prime} \in \mathbb{R}^{d}$ such that the new estimates $\hat{f}{i}^{\prime}$ at the points $x{i}^{\prime} \in \mathbb{R}^{d}$ agree with the original density estimates, i.e.,
$$\hat{f}{i}^{\prime}=\hat{f}{i}, \quad i=1, \ldots, m .$$
For this purpose, one can attempt, for example, to minimize the mean squared deviation of $\hat{f}{i}^{\prime}$ from $\hat{f}{i}$ as a function of the $x_{i}^{\prime}$ s, but such an approach would result in a non-convex optimization problem with many local minima. We formulate an alternative approach involving semidefinite programming, for the special case of the Epanechnikov kernel [25], which is known to be asymptotically optimal for density estimation, and is convenient for formulating a convex optimization problem for the matrix of inner products (the Gram matrix, or the kernel matrix) of the low dimensional data set $X^{\prime}$.

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

The Epanechnikov kernel. The Epanechnikov kernel $k_{e}$ in $d$ dimensions is defined as,
$$k_{e}\left(\left|x_{i}-x_{j}\right|\right)=\left{\begin{array}{cc} N_{e}\left(1-\left|x_{i}-x_{j}\right|^{2}\right), & 0 \leq\left|x_{i}-x_{j}\right| \leq 1 \ 0, & 1 \leq\left|x_{i}-x_{j}\right| \end{array}\right.$$
where $N_{e}$ is the normalization constant that ensures $\int_{\mathbb{R}{d}} k{e}\left(\left|x-x^{\prime}\right|\right) d^{d} x^{\prime}=1$. We will assume that the kernel used in the estimates $\hat{f}{i}$ and $\hat{f}{i}^{\prime}$ of the density via (3.7) is the Epanechnikov kernel. Owing to its quadratic form (3.9), this kernel facilitates the formulation of a convex optimization problem. Instead of seeking the dimensionally reduced version $X^{\prime}=\left{x_{1}^{\prime}, \ldots, x_{n}^{\prime}\right}$ of the data set directly, we will first aim to obtain the kernel matrix $K_{i j}=x_{i}^{\prime} \cdot x_{j}^{\prime}$ for the low-dimensional data points. This is a common approach in the manifold learning literature, where one obtains the low-dimensional data points themselves from the $K_{i j}$ via a singular value decomposition.

We next formulate the DPM optimization problem using the Epanechnikov kernel, and comment on the motivation behind it. As in the case of distance-based manifold learning methods, there will likely be various approaches to density-preserving dimensional reduction, some computationally more efficient than the one discussed here. We hope the discussions in this chapter will stimulate further research in this area.

Given the estimated densities $\hat{f}{i}$, we seek a symmetric, positive semidefinite inner product matrix $K{i j}=x_{i}^{\prime} \cdot x_{j}^{\prime}$ that results in $d$-dimensional density estimates that agree with $\hat{f}_{i}$. In order to deal with the non-uniqueness problem mentioned during our discussion of densitypreserving maps between manifolds (which likely carries over to the discrete setting), we need to pick a suitable objective function to maximize. We choose the objective function to be the same as that of Maximum Variance Unfolding (MVU) [29], namely, $\operatorname{trace}(K)$. After getting rid of translations by constraining the center of mass of the dimensionally reduced data points to the origin, maximizing the objective function trace $(K)$ becomes equivalent to maximizing the sum of the squared distances between the data points [29].

While the objective function for DPM is the same as that of MVU, the constraints of the former will be weaker. Instead of preserving the distances between $k$-nearest neighbors, the DPM optimization defined below preserves the total contribution of the original $k$-nearest neighbors to the density estimate at the data points. As opposed to MVU, this allows for local stretches of the data set, and results in optimal kernel matrices $K$ that can be faithfully represented by a smaller number of dimensions than the intrinsic dimensionality suggested by MVU. For instance, while MVU is capable of unrolling data on the Swiss roll onto a flat plane, it is impossible to lay data from a spherical cap onto the plane while keeping the distances to the $k$ th nearest neighbors fixed. ${ }^{11}$ Thus, the constraints of the optimization in MVU are too stringent to give an inner product matrix $K$ of rank 2, when the original data is on an intrinsically curved surface in $\mathbb{R}^{3}$. We will see below that the looser constraints of DPM allow it to do a better job in capturing the intrinsic dimensionality of a curved surface.

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

In this chapter, we discussed density preserving maps, a density-based alternative to distancebased methods of manifold learning. This method aims to perform dimensionality reduction on large-dimensional data sets in a way that preservs their density. By using a classical result due to Moser, we proved that density preserving maps to $\mathbb{R}^{d}$ exist even for data on intrinsically curved $d$-dimensional submanifolds of $\mathbb{R}^{D}$ that are globally, or topologically “simple.” Since the underlying probability density function is arguably one of the most fundamental statistical quantities pertaining to a data set, a method that preserves densities while performing dimensionality reduction is guaranteed to preserve much valuable structure in the data. While distance-preserving approaches distort data on intrinsically curved spaces in various ways, density preserving maps guarantee that certain fundamental statistical information is conserved.

We reviewed a method of estimating the density on a submanifold of Euclidean space. This method was a slightly modified version of the classical method of kernel density estimation, with the additional property that the convergence rate was determined by the intrinsic dimensionality of the data, instead of the full dimensionality of the Euclidean space the data was embedded in. We made a further modification on this estimator to allow for variable “bandwidths,” and used it with a specific kernel function to set up a semidefinite optimization problem for a proof-of-concept approach to density preserving maps. The objective function used was identical to the one in Maximum Variance Unfolding [29], but the constraints were significantly weaker than the distance-preserving constraints in MVU. By testing the methods on two relatively small, synthetic data sets, we experimentally confirmed the theoretical expectations and showed that density preserving maps are better in detecting and reducing to the intrinsic dimensionality of the data than some of the commonly used distance-based approaches that also work by first estimating a kernel matrix.
While the initial formulation presented in this chapter is not yet scalable to large data sets, we hope our discussion will motivate our readers to pursue the idea of density preserving maps further, and explore alternative, superior formulations. One possible approach to speeding up the computation is to use fast semidefinite programming techniques [4].

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

Fj^=F^(Xj)=1米∑一世1H一世jdķd(|Xj−X一世|DH一世j).

F^一世′=F^一世,一世=1,…,米.

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

Epanechnikov 内核。Epanechnikov 内核ķ和在d维度定义为
$$k_{e}\left(\left|x_{i}-x_{j}\right|\right)=\left{ñ和(1−|X一世−Xj|2),0≤|X一世−Xj|≤1 0,1≤|X一世−Xj|\对。$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|流形学习代写manifold data learning代考| Density Estimation on Submanifolds

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Introduction

Kernel density estimation (KDE) [21] is one of the most popular methods of estimating the underlying probability density function (PDF) of a data set. Roughly speaking, KDE consists of having the data points contribute to the estimate at a given point according to their distances from that point – closer the point, the bigger the contribution. More precisely, in the simplest multi-dimensional KDE [5], the estimate $\hat{f}{m}\left(\mathbf{y}{0}\right)$ of a PDF $f\left(\mathbf{y}{0}\right)$ at a point $\mathbf{y}{0} \in \mathbb{R}^{D}$ is given in terms of a sample $\left{\mathbf{y}{1}, \ldots, \mathbf{y}{m}\right}$ as,
$$\hat{f}{m}\left(\mathbf{y}{0}\right)=\frac{1}{m} \sum_{i=1}^{m} \frac{1}{h_{m}^{D}} K\left(\frac{\left|\mathbf{y}{i}-\mathbf{y}{0}\right|}{h_{m}}\right),$$
where $h_{m}>0$, the bandwidth, is chosen to approach to zero in a suitable manner as the number $m$ of data points increases, and $K:[0, \infty) \rightarrow[0, \infty)$ is a kernel function that satisfies certain properties such as boundedness. Various theorems exist on the different types and rates of convergence of the estimator to the correct result. The earliest result on the pointwise convergence rate in the multivariable case seems to be given in [5], where it is stated that under certain conditions for $f$ and $K$, assuming $h_{m} \rightarrow 0$ and $m h_{m}^{D} \rightarrow \infty$ as $m \rightarrow \infty$, the mean squared error in the estimate $\hat{f}\left(\mathbf{y}{0}\right)$ of the density at a point goes to zero with the rate, $$\operatorname{MSE}\left[\hat{f}{m}\left(\mathbf{y}{0}\right)\right]=\mathrm{E}\left[\left(\hat{f}{m}\left(\mathbf{y}{0}\right)-f\left(\mathbf{y}{0}\right)\right)^{2}\right]=O\left(h_{m}^{4}+\frac{1}{m h_{m}^{D}}\right)$$
as $m \rightarrow \infty$. If $h_{m}$ is chosen to be proportional to $m^{-1 /(D+4)}$, one gets,
$$\operatorname{MSE}\left[\hat{f}{m}(p)\right]=O\left(\frac{1}{m^{4 /(D+4)}}\right),$$ as $m \rightarrow \infty$. The two conditions $h{m} \rightarrow 0$ and $m h_{m}^{D} \rightarrow \infty$ ensure that, as the number of data points increases, the density estimate at a point is determined by the values of the density in a smaller and smaller region around that point, but the number of data points contributing to the estimate (which is roughly proportional to the volume of a region of size $h_{m}$ ) grows unboundedly, respectively.

## 机器学习代写|流形学习代写manifold data learning代考|Motivation for the Submanifold Estimator

We would like to estimate the values of a PDF that lives on an (unknown) $d$-dimensional Riemannian submanifold $M$ of $\mathbb{R}^{D}$, where $d<D$. Usually, $D$-dimensional KDE does not work for such a distribution. This can be intuitively understood by considering a distribution on a line in the plane: 1-dimensional KDE performed on the line (with a bandwidth $h_{m}$ satisfying the asymptotics given above) would converge to the correct density on the line, but 2-dimensional KDE, differing from the former only by a normalization factor that blows up as the bandwidth $h_{m} \rightarrow 0$ (compare (3.1) for the cases $D=2$ and $D=1$ ), diverges. This behavior is due to the fact that, similar to a “delta function” distribution on $\mathbb{R}$, the $D$-dimensional density of a distribution on a $d$-dimensional submanifold of $\mathbb{R}^{D}$ is, strictly speaking, undefined – the density is zero outside the submanifold, and in order to have proper normalization, it has to be infinite on the submanifold. More formally, the $D$ dimensional probability measure for a $d$-dimensional PDF supported on $M$ is not absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{D}$, and does not have a probability

density function on $\mathbb{R}^{D}$. If one attempts to use $D$-dimensional KDE for data drawn from such a probability measure, the estimator will “attempt to converge” to a singular PDF; one that is infinite on $M$, zero outside.

For a distribution with support on a line in the plane, we can resort to 1-dimensional KDE to get the correct density on the line, but how could one estimate the density on an unknown, possibly curved submanifold of dimension $d<D$ ? Essentially the same approach works: even for data that lives on an unknown, curved $d$-dimensional submanifold of $\mathbb{R}^{D}$, it suffices to use the $d$-dimensional kernel density estimator with the Euclidean distance on $\mathbb{R}^{D}$ to get a consistent estimator of the submanifold density. Furthermore, the convergence rate of this estimator can be bounded as in (3.3), with $D$ being replaced by $d$, the intrinsic dimension of the submanifold. [20]

The intuition behind this approach is based on three facts: 1) For small bandwidths, the main contribution to the density estimate at a point comes from data points that are nearby; 2) For small distances, a $d$-dimensional Riemannian manifold “looks like” $\mathbb{R}^{d}$, and densities in $\mathbb{R}^{d}$ should be estimated by a $d$-dimensional kernel, instead of a $D$-dimensional one; and 3) For points of $M$ that are close to each other, the intrinsic distances as measured on $M$ are close to Euclidean distances as measured in the surrounding $\mathbb{R}^{D}$. Thus, as the number of data points increases and the bandwidth is taken to be smaller and smaller, estimating the density by using a kernel normalized for $d$ dimensions and distances as measured in $\mathbb{R}^{D}$ should give a result closer and closer to the correct value.

We will next give the formal definition of the estimator motivated by these considerations, and state the theorem on its asymptotics. As in the original work of Parzen [21], the pointwise consistence of the estimator can be proven by using a bias-variance decomposition. The asymptotic unbiasedness of the estimator follows from the fact that as the bandwidth converges to zero, the kernel function becomes a “delta function.” Using this fact, it is possible to show that with an appropriate choice for the vanishing rate of the bandwidth, the variance also vanishes asymptotically, completing the proof of the pointwise consistency of the estimator.

## 机器学习代写|流形学习代写manifold data learning代考|Statement of the Theorem

Let $(M, \mathbf{g})$ be a $d$-dimensional, embedded, complete, compact Riemannian submanifold of $\mathbb{R}^{D}(d0 .^{7}$ Let $d(p, q)=d_{p}(q)$ be the length of a length-minimizing geodesic in $M$ between $p, q \in M$, and let $u(p, q)=u_{p}(q)$ be the geodesic distance between $p$ and $q$ as measured in $\mathbb{R}^{D}$ (thus, $u(p, q)$ is simply the Euclidean distance between $p$ and $q$ in $\left.\mathbb{R}^{D}\right)$. Note that $u(p, q) \leq d(p, q)$. We will denote the Riemannian volume measure on $M$ by $V$, and the volume form by $d V .^{8}$
Theorem 3.3.1 Let $f: M \rightarrow[0, \infty)$ be a probability density function defined on $M$ (so that the related probability measure is $f V)$, and $K:[0, \infty) \rightarrow[0, \infty)$ be a continuous function that vanishes outside $[0,1)$, is differentiable with a bounded derivative in $[0,1)$, and satisfies the normalization condition, $\int_{|\mathbf{z}| \leq 1} K(|\mathbf{z}|) d^{d} \mathbf{z}=1$. Assume $f$ is differentiable to second order in a neighborhood of $p \in M$, and for a sample $q_{1}, \ldots, q_{m}$ of size $m$ drawn from the

density $f$, define an estimator $\hat{f}{m}(p)$ of $f(p)$ as, $$\hat{f}{m}(p)=\frac{1}{m} \sum_{j=1}^{m} \frac{1}{h_{m}^{d}} K\left(\frac{u_{p}\left(q_{j}\right)}{h_{m}}\right)$$
where $h_{m}>0$. If $h_{m}$ satisfies $\lim {m \rightarrow \infty} h{m}=0$ and $\lim {m \rightarrow \infty} m h{m}^{d}=\infty$, then, there exist non-negative numbers $m_{}, C_{b}$, and $C_{V}$ such that for all $m>m_{}$ the mean squared error of the estimator (3.4) satisfies,
$$\operatorname{MSE}\left[\hat{f}{m}(p)\right]=\mathrm{E}\left[\left(\hat{f}{m}(p)-f(p)\right)^{2}\right]<C_{b} h_{m}^{4}+\frac{C_{V}}{m h_{m}^{d}}$$
If $h_{m}$ is chosen to be proportional to $m^{-1 /(d+4)}$, this gives,
$$\mathrm{E}\left[\left(f_{m}(p)-f(p)\right)^{2}\right]=O\left(\frac{1}{m^{4 /(d+4)}}\right)$$
as $m \rightarrow \infty$.
Thus, the bound on the convergence rate of the submanifold density estimator is as in (3.2), (3.3), with the dimensionality $D$ replaced by the intrinsic dimension $d$ of $M$. As mentioned above, the proof of this theorem follows from two lemmas on the convergence rates of the bias and the variance; the $h_{m}^{4}$ term in the bound corresponds to the bias, and the $1 / m h_{m}^{d}$ term corresponds to the variance; see $[20]$ for details. This approach to submanifold density estimation was previously mentioned in [11], and the thesis [10] contains the details, although in a more technical and general approach than the elementary one followed in [20].

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

F^米(是0)=1米∑一世=1米1H米Dķ(|是一世−是0|H米),

MSE⁡[F^米(p)]=这(1米4/(D+4)),作为米→∞. 两个条件H米→0和米H米D→∞确保随着数据点数量的增加，一个点的密度估计值由该点周围越来越小的区域中的密度值决定，但对估计值有贡献的数据点数量（大致成正比）到一个大小区域的体积H米) 分别无限增长。

## 机器学习代写|流形学习代写manifold data learning代考|Statement of the Theorem

MSE⁡[F^米(p)]=和[(F^米(p)−F(p))2]<CbH米4+C在米H米d

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|流形学习代写manifold data learning代考|The Existence of Density Preserving Maps

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Moser’s Theorem

Riemannian manifolds. We begin by restricting our attention to data subspaces which are Riemannian submanifolds of $\mathbb{R}^{D}$. Riemannian manifolds provide a generalization of the notion of a smooth surface in $\mathbb{R}^{3}$ to higher dimensions. As first clarified by Gauss in the two-dimensional case and by Riemann in the general case, it turns out that intrinsic features of the geometry of a surface, such as the lengths of its curves or intrinsic distances between its points, etc., can be given in terms of the so-called metric tensor ${ }^{2} \mathrm{~g}$, without referring to the particular way the the surface is embedded in $\mathbb{R}^{3}$. A space whose geometry is defined in terms of a metric tensor is called a Riemannian manifold (for a rigorous definition, see, e.g., $[12,16,2])$.

The Gauss/Riemann result mentioned above states that if the intrinsic curvature of a Riemannian manifold $\left(M, \mathbf{g}_{M}\right)$ is not zero in an open set $U \in M$, it is not possible to find

a map from $M$ into $\mathbb{R}^{d}$ that preserves the distances between the points of $U$. Thus, there exists a local obstruction, namely, the curvature, to the existence of distance-preserving maps. It turns out that no such local obstruction exists for volume-preserving maps. The only invariant is a global one, namely, the total volume. ${ }^{3}$ This is the content of Moser’s theorem on volume-preserving maps, which we state next.

Theorem 3.2.1 (Moser [18]) Let $\left(M, \mathbf{g}{M}\right)$ and $\left(N, \mathbf{g}{N}\right)$ be two closed, connected, orientable, $d$-dimensional differentiable manifolds that are diffeomorphic to each other. Let $\tau_{M}$ and $\tau_{N}$ be volume forms, i.e., nowhere vanishing $d$-forms on these manifolds, satisfying $\int_{M} \tau_{M}=$ $\int_{N} \tau_{N}$. Then, there exists a diffeomorphism $\phi: M \rightarrow N$ such that $\tau_{M}=\phi^{*} \tau_{N}$, i.e., the volume form on $M$ is the same as the pull-back of the volume form on $N$ by $\phi .^{4}$

The meaning of this result is that, if two manifolds with the same “global shape” (i.e., two manifolds that are diffeomorphic) have the same total volume, one can find a map between them that preserves the volume locally. The surfaces of a mug and a torus are the classical examples used for describing global, topological equivalence. Although these objects have the same “global shape” (topology/smooth structure) their intrinsic, local geometries are different. Moser’s theorem states that if their total surface areas are the same, one can find a map between them that preserves the areas locally, as well, i.e., a map that sends all small regions on one surface to regions in the other surface in a way that preserves the areas.

Using this theorem, we now show that it is possible to find density-preserving maps between Riemannian manifolds that have the same total volume. This is due to the fact that if local volumes are preserved under a map, the density of a distribution will also be preserved.

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

These results were formulated in terms of so-called closed manifolds, i.e., compact manifolds without boundary. The practical dimensionality reduction problem we would like to address, on the other hand, involves starting with a $d$-dimensional data submanifold $M$ of $\mathbb{R}^{D}$ (where $d<D)$, and dimensionally reducing to $\mathbb{R}^{d}$. In order to be able to do this diffeomorphically, $M$ must be diffeomorphic to a subspace of $\mathbb{R}^{d}$, which is not generally the case for closed manifolds. For instance, although we can find a diffeomorphism from a hemisphere (a manifold with boundary, not a closed manifold) into the plane, we cannot find one from the unit sphere (a closed manifold) into the plane. This is a constraint on all dimensional reduction algorithms that preserve the global topology of the data space, not just density preserving maps. Any algorithm that aims to avoid “tearing” or “folding” the data subspace during the reduction will fail on problems like reducing a sphere to $\mathbb{R}^{2.5}$

Thus, in order to show that density preserving maps into $\mathbb{R}^{d}$ exist for a useful class of $d$-dimensional data manifolds, we have to make sure that the conclusion of Moser’s theorem and our corollary work for certain manifolds with boundary, or for certain non-compact manifolds, as well. Fortunately, this is not so hard, at least for a simple class of manifolds that is enough to be useful. In proving his theorem for closed manifolds, Moser [18] first gives a proof for a single “coordinate patch” in such a manifold, which, basically, defines a compact manifold with boundary minus the boundary itself. Not all $d$-dimensional manifolds with boundary (minus their boundaries) can be given by atlases consisting of a single coordinate patch, but the ones that can be so given cover a wide range of curved Riemannian manifolds, including the hemisphere and the Swiss roll, possibly with punctures. In the following, we will assume that $M$ consists of a single coordinate patch.

## 机器学习代写|流形学习代写manifold data learning代考|Intuition on Non-Uniqueness

Note that the results above claim the existence of volume (or density) preserving maps, but not uniqueness. In fact, the space of volume-preserving maps is very large. An intuitive way to see this is to consider the flow of an incompressible fluid in $\mathbb{R}^{3}$. The fluid may cover the same region in space at two given times, but the fluid particles may have gone through significant shuffling. The map from the original configuration of the fluid to the final one is a volume preserving diffeomorphism, assuming the flow is smooth. The infinity of ways a fluid can move shows the infinity of ways of preserving volume.

Distance-preserving maps may also have some non-uniqueness, but this is parametrized by a finite-dimensional group, namely, the isometry group of the Riemannian manifold under consideration. ${ }^{6}$ The case of volume-preserving maps is much worse, the space of volumepreserving diffeomorphisms being infinite-dimensional. Since the aim of this chapter is to describe a manifold-learning method that preserves volumes/densities, we are faced with the following question: Given a data manifold with intrinsic dimension $d$ that is diffeomorphic to a subset of $\mathbb{R}^{d}$, which map, in the infinite-dimensional space of volume-preserving maps from this manifold to $\mathbb{R}^{d}$, is the “best”? In Section 3.4, we will describe an approach to this problem by setting up a specific optimization procedure. But first, let us describe a method for estimating densities on submanifolds.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|LEM Results

Figures $2.5-2.7$ show the results after using LEM for different values of $k$. As the value of $k$ increases from 1 to higher values we notice the spreading of the embedded data. The bottom subplot shows the nearest neighbor graph with $k=1$ as shown in Figure 2.7. The right plot shows the embedding of the graph. It is interesting to observe how the embedded data loses its local neighborhood information. The embedding practically happens along the second principal eigenvector (the first being Zero Vector). As the value of $k$ is increased to 2, we observe that embedding happens along the second and third principal axes. See Figure 2.7. For $k=1$ the graph is highly disconnected and for $k=2$ the graphs has much less isolated pieces of graphs. One interesting thing to observe is that as the connectivity of the graph increases the low-dimensional representation begins to preserve the local information.
The graph with $k=2$ and its embedding is shown in Figure 2.8. Increasing the neighborhood information to 2 neighbors is still not able to represent the continuity of the original manifold. Figure $2.7$ shows the graph with $k=3$ and its embedding. Increasing the neighborhood information to 3 neighbors better represents the continuity of the original manifold. Figure $2.5$ shows the graph with $k=5$ and its embedding. Increasing the neighborhood information to 5 neighbors better represents the continuity of the original manifold. Similar results are obtained by increasing the the number of neighbors, however, it should be noted that when the number of neighbors is very high then the graph starts to get influenced by ambient neigbhors.

We see similar results for the face images. The three plots in Figure $2.6$ show the embedding results obtained using LEM when the neighborhood graphs are created using $k=1, k=2$, and $k=5$. The top and the middle plot validate the limitation of LEM.

## 机器学习代写|流形学习代写manifold data learning代考|Bibliographical and Historical Remarks

Dimensionality reduction is an important research area in data analysis with an extensive research literature. Both linear and non-linear methods exist, and each category has both supervised and unsupervised versions. In this section we will briefly mention some of the salient works that have been proposed in the area of locally preserving manifold learning: see $[8]$ for a broader survey.

Lee and Seung [12] showed that many high dimensional data such as a series of related images, video frames, etc. lie on a much lower-dimensional manifold instead of being scattered throughout the feature space. This particular observation has motivated researchers to develop dimension reduction algorithms that try to learn an embedded manifold in a high-dimensional space.

ISOMAP [14] learns the manifold by exploring geodesic distances. In fact the algorithm tries to preserve the geometry of the data on the manifold by noting the points in the neighborhood of each point. The algorithm is defined as such:

1. Form a neighborhood graph $G$ for the dataset, based, for instance, on the $K$ nearest neighbors of each point $x_{i}$ –
2. For every pair of nodes in the graph, compute the shortest path, using Dijkstras algorithm, as an estimate of intrinsic distance on the data manifold. The weights of edges of the graphs are computed based on the Euclidean distance measure.
3. Classical Multi-Dimensional Scaling algorithm is computed using these pairwise distances to find a lower dimensional embedding $y_{i}$.

Bernstein et al. [22] have described the convergence properties of the estimation procedure for the intrinsic distances. For large and dense data sets, computation of pairwise distances is time consuming, and moreover the calculation of eigenvalues can be computationally intensive for large data sets. Such constraints have motivated researchers to find simpler variations of the Isomap algorithm. One such algorithm uses subsampled data called landmarks. Firstly, it calculates Isomap for random points called landmarks and between those landmarks a simple triangulation algorithm is applied.

Locally Linear Embedding (LLE) is an unsupervised learning method based on global and local optimization [11]. It is is similar to Isomap in the sense that it generates a graphical representation of the data set. However, it is different from Isomap as it only attempts to preserve local structures of the data. Because of the locality property used in LLE, the algorithm allows for successful embedding of nonconvex manifolds. An important point to be noted is that LLE creates the local properties of a manifold using the linear combinations of $k$ nearest neighbors of the data $x_{i}$. LLE attempts to create a local regression like model and thereby tries to fit a hyperplane through the data point $x_{i}$. This appears to be reasonable for smooth manifolds where the nearest neighbors align themselves well in a linear space. For very non-smooth or noisy data sets, LLE does not perform well. It has been noted that LLE preserves the reconstruction weights in the space of lower dimensionality, as the reconstruction weights of a data point are invariant to linear transformational operations like translation, rotation, etc.

## 机器学习代写|流形学习代写manifold data learning代考|Arkadas Ozakin, Nikolaos Vasiloglou II, Alexander Gray

Much of the recent work in manifold learning and nonlinear dimensionality reduction focuses on distance-based methods, i.e., methods that aim to preserve the local or global (geodesic) distances between data points on a submanifold of Euclidean space. While this is a promising approach when the data manifold is known to have no intrinsic curvature (which is the case for common examples such as the “Swiss roll”), classical results in Riemannian geometry show that it is impossible to map a $d$-dimensional data manifold with intrinsic curvature into $\mathbb{R}^{d}$ in a manner that preserves distances. Consequently, distance-based methods of dimensionality reduction distort intrinsically curved data spaces, and they often do so in unpredictable ways. In this chapter, we discuss an alternative paradigm of manifold learning. We show that it is possible to perform nonlinear dimensionality reduction by preserving the underlying density of the data, for a much larger class of data manifolds than intrinsically flat ones, and demonstrate a proof-of-concept algorithm demonstrating the promise of this approach.

Visual inspection of data after dimensional reduction to two or three dimensions is among the most common uses of manifold learning and nonlinear dimensionality reduction. Typically, what is sought by the user’s eye in two or three-dimensional plots is clustering and other relationships in the data. Knowledge of the density, in principle, allows one to identify such basic structures as clusters and outliers, and even define nonparametric classifiers; the underlying density of a data set is arguably one of the most fundamental statistical objects that describe it. Thus, a method of dimensionality reduction that is guaranteed to preserve densities may well be preferable to methods that aim to preserve distances, but end up distorting them in uncontrolled ways.

Many of the manifold learning methods require the user to set a neighborhood radius $h$, or, for $k$-nearest neighbor approaches, a positive integer $k$, to be used in determining the neighborhood graph. Most of the time, there is no automatic way to pick the appropriate values of the tweak parameters $h$ and $k$, and one resorts to trial and error, looking for values that result in reasonable-looking plots. Kernel density estimation, one of the most popular and useful methods of estimating the underlying density of a data set, comes with a natural way to choose $h$ or $k$; it suggests to us to pick the value that maximizes a cross-validation score for the density estimate. While the usual kernel density estimation does not allow one to estimate the density of data on submanifolds of Euclidean space, a small modification

allows one to do so. This modification and its ramifications are discussed below in the context of density-preserving maps.

The chapter is organized as follows. In Section 3.2, using a theorem of Moser, we prove the existence of density preserving maps into $\mathbb{R}^{d}$ for a large class of $d$-dimensional manifolds, and give an intuitive discussion on the nonuniqueness of such maps. In Section 3.3, we describe a method for estimating the underlying density of a data set on a Riemannian submanifold of Euclidean space. We state the main result on the consistency of this submanifold density estimator, and give a bound on its convergence rate, showing that the latter is determined by the intrinsic dimensionality of the data instead of the full dimensionality of the feature space. This, incidentally, shows that the curse of dimensionality in the widely-used method of kernel density estimation is not as severe as is generally believed, if the method is properly modified for data on submanifolds. In Section 3.4, using a modified version of the estimator defined in Section 3.3, we describe a proof-of-concept algorithm for density preserving maps based on semidefinite programming, and give experimental results. Finally, in Sections $7.7$ and 3.6, we summarize the chapter and discuss relevant bibliography.

## 机器学习代写|流形学习代写manifold data learning代考|Bibliographical and Historical Remarks

Lee 和 Seung [12] 表明，许多高维数据（例如一系列相关图像、视频帧等）位于低得多的流形上，而不是分散在整个特征空间中。这一特殊观察促使研究人员开发降维算法，试图在高维空间中学习嵌入式流形。

ISOMAP [14] 通过探索测地距离来学习流形。事实上，该算法试图通过注意每个点附近的点来保留流形上数据的几何形状。算法定义如下：

1. 形成邻域图G对于数据集，例如，基于ķ每个点的最近邻X一世 –
2. 对于图中的每一对节点，使用 Dijkstras 算法计算最短路径，作为数据流形上内在距离的估计。图的边权重是根据欧几里得距离度量计算的。
3. 使用这些成对距离计算经典的多维缩放算法以找到较低维的嵌入是一世.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Shounak Roychowdhury and Joydeep Ghosh

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|Graph Laplacian

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.

## 机器学习代写|流形学习代写manifold data learning代考|Global Information of Manifold

Global information has not been used in manifold learning since it is widely believed that global information may capture unnecessary data (like ambient data points) that should be avoided when dealing with manifolds.

However, some recent research results show that that it might be useful to to explore global information in a more constrained manner for manifold learning. Costa and Hero show that it is possible to use a Geodesic Minimum Spanning Tree (GMST) on the manifold to estimate the intrinsic dimension and intrinsic entropy of the manifold [4].

Costa and Hero showed in the following theorem that is possible to learn the intrinsic entropy and intrinsic dimension of a non-linear manifold by extending the BHH theorem [1], a well-known result in Geometric Probability.

Theorem: [Generalization of BHH Theorem to Embedded manifolds: [4]] Let $\mathcal{M}$ be a smooth compact $m$-dimensional manifold embedded in $\mathbb{R}^{d}$ through the diffeomorphism $\phi: \Omega \rightarrow \mathcal{M}$, and $\Omega \in \mathbb{R}^{d}$. Assume $2 \leq m \leq d$ and $0<\gamma<m$. Suppose that $Y_{1}, Y_{2}, \ldots$ are iid random vectors on $\mathcal{M}$ having a common density function $f$ with respect to a Lebesgue measure $\mu_{\mathcal{M}}$ on $\mathcal{M}$. Then the length functional $T_{\gamma}^{\mathbb{R}^{m}} \phi_{-1}\left(Y_{n}\right)$ of the MST spanning $\phi^{-1}\left(Y_{n}\right)$ satisfies the equation shown below in an almost sure sense:

$$\lim {n \rightarrow \infty} \frac{T{\gamma}^{2^{m}} \phi_{-1}\left(Y_{n}\right)}{n \frac{(d-1)}{d}}=$$
where $\alpha=(m-\gamma) / m$, and is always between $0<\alpha<1, J$ is the Jacobian, and $\beta_{m}$ is $a$ constant which depends on $m$.

Based on the above theorem we use MST on the entire data set as a source of global information. For more details see $[4]$, and more background information see [15] and [13].
The basic principle of GLEM is quite straightforward. The objective function that is to be minimized is given by the following (it is has the same flavor and notation used in [2]):
\begin{aligned} & \sum_{i, j}\left|\mathbf{y}^{(\mathbf{i})}-\mathbf{y}^{(\mathbf{j})}\right|_{2}^{2}\left(W_{i j}^{N N}+W_{i j}^{M S T}\right) \ =& \operatorname{tr}\left(\mathbf{Y}^{T} L\left(G_{N N}\right) \mathbf{Y}+\mathbf{Y}^{T} L\left(G_{M S T}\right) \mathbf{Y}\right) \ =& \operatorname{tr}\left(\mathbf{Y}^{T}\left(L\left(G_{N N}\right)+L\left(G_{M S T}\right)\right) \mathbf{Y}\right) \ =& \operatorname{tr}\left(\mathbf{Y}^{T} L(J) \mathbf{Y}\right) . \end{aligned}
where $\mathbf{y}^{(i)}=\left[y_{1}(i), \ldots, y_{m}(i)\right]^{T}$, and $m$ is the dimension of embedding. $W_{i j}^{N N}$ and $W_{i j}^{M S T}$ are weighted matrices of k-Nearest Neighbor graph and the MST graph respectively. In other words, we have
$$\operatorname{argmin}{\mathbf{Y T} \mathbf{Y}=\mathbf{I}} \mathbf{Y}^{T} L \mathbf{Y}$$ such that $Y=\left[\mathbf{y}{\mathbf{1}}, \mathbf{y}{\mathbf{2}}, \ldots, \mathbf{y}{\mathbf{m}}\right]$ and $\mathbf{y}^{(\mathbf{i})}$ is the $m$-dimensional representation of $i^{\text {th }}$ vertex. The solutions to this optimization problem are the eigenvectors of the generalized eigenvalue problem
$$L \mathbf{Y}=\Lambda D \mathbf{Y}$$
The GLEM algorithm is described in Algorithm $1 .$

## 机器学习代写|流形学习代写manifold data learning代考|Shounak Roychowdhury and Joydeep Ghosh

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|Graph Laplacian

Fiedler [6] 将图的拉普拉斯算子定义为正则图的对称矩阵，其中一种是一个邻接矩阵 (一种吨是邻接矩阵的转置），一世是单位矩阵，并且n是正则图的度数：

Chung 的定义（见 [3]）——下面给出——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和在，加权邻接矩阵。

## 机器学习代写|流形学习代写manifold data learning代考|Global Information of Manifold

Costa 和 Hero 在以下定理中表明，可以通过扩展 BHH 定理 [1] 来学习非线性流形的内在熵和内在维数，这是几何概率中的一个众所周知的结果。

GLEM 的基本原理非常简单。要最小化的目标函数由以下给出（它与 [2] 中使用的风格和符号相同）：
∑一世,j|是(一世)−是(j)|22(在一世jññ+在一世j米小号吨) =tr⁡(是吨大号(Gññ)是+是吨大号(G米小号吨)是) =tr⁡(是吨(大号(Gññ)+大号(G米小号吨))是) =tr⁡(是吨大号(Ĵ)是).

GLEM 算法在算法中描述1.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Diffusion Maps

The basic idea of Diffusion MaPs (Nadler, Lafon, Coifman, and Kevrekidis, 2005; Coifman and Lafon, 2006) uses a Markov chain constructed over a graph of the data points, followed by an eigenanalysis of the probability transition matrix of the Markov chain. As with the other algorithms in this Section, there are three steps in this algorithm, with the first and second steps the same as for Laplacian eigenmaps. Although a nearest-neighbor search (Step 1) was not explicitly considered in the above papers on diffusion maps as a means of constructing the graph (Step 2), a nearest-neighbor search is included in software packages for computing diffusion maps. For an example in astronomy of a diffusion map incorporating a nearest-neighbor search, see Freeman, Newman, Lee, Richards, and Schafer (2009).

1. Nearest-Neighbor Search. Fix an integer $K$ or an $\epsilon>0$. Define a $K$-neighborhood $N_{i}^{K}$ or an $\epsilon$-neighborhood $N_{i}^{e}$ of the point $\mathbf{x}{i}$ as in Step 1 of Laplacian eigenmaps. In general, let $N{i}$ denote the neighborhood of $\mathbf{x}_{i}$.Pairwise Adjacency Matrix. The $n$ data points $\left{\mathbf{x}{i}\right}$ in $\Re^{r}$ can be regarded as a graph $\mathcal{G}=\mathcal{G}(\mathcal{V}, \mathcal{E})$ with the data points playing the role of vertices $\mathcal{V}=\left{\mathbf{x}{1}, \ldots, \mathbf{x}{n}\right}$, and the set of edges $\mathcal{E}$ are the connection strengths (or weights), $w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)$, between pairs of adjacent vertices, $$w{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{i}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { otherwise. }\end{cases} 2.$$
3. This is a Gaussian kernel with width $\sigma$; however, other kernels may be used. Kernels such as (1.52) ensure that the closer two points are to each other, the larger the value of $w$. For convenience in exposition, we will suppress the fact that the elements of most of the matrices depend upon the value of $\sigma$. Then, $\mathbf{W}=\left(w_{i j}\right)$ is a pairwise adjacency matrix between the $n$ points. To make the matrix $\mathbf{W}$ even more sparse, values of its entries that are smaller than some given threshold (i.e., the points in question are far apart from each other) can be set to zero. The graph $\mathcal{G}$ with weight matrix W gives information on the local geometry of the data.
4. Spectral embedding. Define $\mathbf{D}=\left(d_{i j}\right)$ to be a diagonal matrix formed from the matrix W by setting the diagonal elements, $d_{i i}=\sum_{j} w_{i j}$, to be the column sums of $\mathbf{W}$ and the off-diagonal elements to be zero. The $(n \times n)$ symmetric matrix $\mathbf{L}=\mathbf{D}-\mathbf{W}$ is the graph Laplacian for the graph $\mathcal{G}$. We are interested in the solutions of the generalized eigenequation, $\mathbf{L v}=\lambda \mathbf{D v}$, or, equivalently, of the matrix
5. $$6. \mathbf{P}=\mathbf{D}^{-1 / 2} \mathbf{L} \mathbf{D}^{-1 / 2}=\mathbf{I}_{n}-\mathbf{D}^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2}, 7.$$
8. which is the normalized graph Laplacian. The matrix $\mathbf{H}=e^{t \mathbf{P}}, t \geq 0$, is usually referred to as the heat kernel. By construction, $\mathbf{P}$ is a stochastic matrix with all row sums equal to one, and, thus, can be interpreted as defining a random walk on the graph $\mathcal{G}$.

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

Recall that, in certain situations, the convexity assumption for IsOMAP may be too restrictive. Instead, we may require that the manifold $\mathcal{M}$ be locally isometric to an open, connected subset of $\Re^{t}$. Popular examples include families of “articulated” images (i.e., translated or rotated images of the same object, possibly through time) that are found in a high-dimensional, digitized-image library (e.g., faces, pictures, handwritten numbers or letters). However, if the pixel elements of each 64 -pixel-by-64-pixel digitized image are represented as a 4,096 -dimensional vector in “pixel space,” it would be very difficult to show that the images really live on a low-dimensional manifold, especially if that image manifold is unknown.

We can model such images using a vector of smoothly varying articulation parameters $\boldsymbol{\theta} \in \boldsymbol{\Theta}$. For example, digitized images of a person’s face that are varied by pose and illumination can be parameterized by two pose parameters (expression [happy, sad, sleepy, surprised, wink] and glasses-no glasses) and a lighting direction (centerlight, leftlight, rightlight, normal); similarly, handwritten ” 2 “s appear to be parameterized essentially by two features, bottom loop and top arch (Tenenbaum, de Silva, and Langford, 2000; Roweis and Saul, 2000). To some extent, learning about an underlying image manifold depends upon whether the images are sufficiently scattered around the manifold and how good is the quality of digitization of each image?

HESSIAN EIGENMAPS (Donoho and Grimes, 2003b) were proposed for recovering manifolds of high-dimensional libraries of articulated images where the convexity assumption is often violated. Let $\Theta \subset \Re^{t}$ be the parameter space and suppose that $\phi: \Theta \rightarrow R^{r}$, where $t<r$. Assume $\mathcal{M}=\phi(\Theta)$ is a smooth manifold of articulated images. The isometry and convexity requirements of IsoMAP are replaced by the following weaker requirements:

• Local Isometry: $\phi$ is a locally isometric embedding of $\Theta$ into $\Re^{r}$. For any point $\mathbf{x}^{\prime}$ in a sufficiently small neighborhood around each point $x$ on the manifold $\mathcal{M}$, the geodesic distance equals the Euclidean distance between their corresponding parameter points $\boldsymbol{\theta}, \boldsymbol{\theta}^{\prime} \in \Theta ;$ that is,
$$d^{M}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|\theta-\theta^{\prime}\right|_{\Theta+}$$
where $\mathbf{x}=\phi(\boldsymbol{\theta})$ and $\mathbf{x}^{\prime}=\phi\left(\boldsymbol{\theta}^{\prime}\right)$
• Connectedness: The parameter space $\theta$ is an open, connected subset of $\Omega^{t}$.
The goal is to recover the parameter vector $\boldsymbol{\theta}$ (up to a rigid motion).

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

Another way of dealing with nonlinear manifold learning is to construct nonlinear versions of linear manifold learning techniques. We have already seen how Isomap provides a nonlinear generalization of MDS. How can we generalize PCA to the nonlinear case? In this Section,

we briefly describe the basic ideas behind POLYNOMIAL PCA, PRINCIPAL CuRVES AND SURFACES, MULTILAYER AUTOASSOCIATIVE NEURAL NETWORKS, and KerneL PCA.
Polynomial PCA
There have been several different attempts to generalize PCA to data living on or near nonlinear manifolds of a lower-dimensional space than input space. The first such idea was to add to the set of $r$ input variables quadratic, cubic, or higher-degree polynomial transformations of those input variables, and then apply linear PCA. The result is POLYNOMIAL PCA (Gnanadesikan and Wilk, 1969), whose embedding coordinates are the eigenvectors corresponding to the smallest few eigenvalues of the expanded covariance matrix.

In the original study of polynomial PCA, the method was illustrated with a quadratic transformation of bivariate input variables. In this scenario, $\left(X_{1}, X_{2}\right)$ expands to become $\left(X_{1}, X_{2}, X_{1}^{2}, X_{2}^{2}, X_{1} X_{2}\right)$. This formulation is feasible, but for larger problems, the possibilities become more complicated. First, the variables in the expanded set will not be scaled in a uniform manner, so that standardization will be necessary, and second, the number of variables in the expanded set will increase rapidly with large $r$, which will lead to bigger computational problems. Gnanadesikan and Wilk’s article, however, gave rise to a variety of attempts to define a more general nonlinear version of PCA.
Principal Curves and Surfaces
The next attempt at creating a nonlinear PCA was PRINCIPAL CURVES AND SURFACES (Hastie, 1984; Hastie and Stuetzle, 1989). A principal curve is a smooth one-dimensional curve that passes through the “middle” of the data, and a principal surface (or principal manifold) is a generalization of a principal curve to a smooth two- or higher-dimensional manifold. So, we can visualize principal curves and surfaces as defining a nonlinear manifold in higher-dimensional input space.

Let $\mathbf{x} \in \Re^{r}$ be a data point and let $\mathbf{f}(\lambda)$ be a curve, $\lambda \in \Lambda$; see Section $1.2 .4$ for definitions. Project $\mathbf{x}$ to a point on $\mathbf{f}(\lambda)$ that is closest in Euclidean distance to $\mathbf{x}$. Define the projection index
$$\lambda_{\mathbf{f}}(\mathbf{x})=\sup {\lambda}\left{\lambda:|\mathbf{x}-\mathbf{f}(\lambda)|=\inf {\mu}|\mathbf{x}-\mathbf{f}(\mu)|\right}$$

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

Diffusion Maps 的基本思想（Nadler、Lafon、Coifman 和 Kevrekidis，2005 年；Coifman 和 Lafon，2006 年）使用在数据点图上构建的马尔可夫链，然后对马尔可夫链的概率转移矩阵进行特征分析. 与本节中的其他算法一样，该算法有三个步骤，第一步和第二步与拉普拉斯特征图相同。尽管在上述关于扩散图的论文中没有明确考虑最近邻搜索（步骤 1）作为构建图的一种手段（步骤 2），但最近邻搜索包含在用于计算扩散图的软件包中。有关包含最近邻搜索的扩散图的天文学示例，请参见 Freeman、Newman、Lee、Richards 和 Schafer (2009)。

1. 最近邻搜索。修复一个整数ķ或一个ε>0. 定义一个ķ-邻里ñ一世ķ或一个ε-邻里ñ一世和点的X一世如拉普拉斯特征图的第 1 步。一般来说，让ñ一世表示邻域X一世.成对邻接矩阵。这n数据点\left{\mathbf{x}{i}\right}\left{\mathbf{x}{i}\right}在ℜr可以看成图G=G(在,和)数据点扮演顶点的角色\mathcal{V}=\left{\mathbf{x}{1}, \ldots, \mathbf{x}{n}\right}\mathcal{V}=\left{\mathbf{x}{1}, \ldots, \mathbf{x}{n}\right}, 和边的集合和是连接强度（或权重），在(X一世,Xj)，在相邻顶点对之间，$$w{ij}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)=\begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{i}\right|^{2}}{2 \sigma^{2} }\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { 否则。}\结束{案例}\begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{i}\right|^{2}}{2 \sigma^{2} }\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { 否则。}\结束{案例} 2.$$
3. 这是一个具有宽度的高斯核σ; 但是，可以使用其他内核。（1.52）等内核保证两点距离越近，值越大在. 为方便说明，我们将隐藏大多数矩阵的元素取决于σ. 然后，在=(在一世j)是之间的成对邻接矩阵n点。制作矩阵在甚至更稀疏，其条目的值小于某个给定阈值（即，所讨论的点彼此相距很远）可以设置为零。图表G权重矩阵 W 给出了数据的局部几何信息。
4. 光谱嵌入。定义D=(d一世j)是通过设置对角元素由矩阵 W 形成的对角矩阵，d一世一世=∑j在一世j, 为的列总和在和非对角线元素为零。这(n×n)对称矩阵大号=D−在是图的拉普拉斯算子G. 我们对广义特征方程的解感兴趣，大号在=λD在, 或者, 等价的, 矩阵
5. $$6. \mathbf{P}=\mathbf{D}^{-1 / 2} \mathbf{L} \mathbf{D}^{-1 / 2}=\mathbf{I}_{n}-\mathbf{D }^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2}, 7.$$
8. 这是归一化图拉普拉斯算子。矩阵H=和吨磷,吨≥0, 通常称为热核。通过施工，磷是一个随机矩阵，所有行和都等于 1，因此可以解释为在图上定义随机游走G.

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

HESSIAN EIGENMAPS (Donoho and Grimes, 2003b) 被提出用于恢复经常违反凸性假设的铰接图像的高维库的流形。让θ⊂ℜ吨是参数空间并假设φ:θ→Rr， 在哪里吨<r. 认为米=φ(θ)是铰接图像的平滑流形。IsoMAP 的等距和凸度要求被以下较弱的要求取代：

• 局部等距：φ是一个局部等距嵌入θ进入ℜr. 对于任何一点X′在每个点周围足够小的邻域中X在歧管上米，测地线距离等于它们对应的参数点之间的欧几里得距离θ,θ′∈θ;那是，
d米(X,X′)=|θ−θ′|θ+
在哪里X=φ(θ)和X′=φ(θ′)
• 连通性：参数空间θ是一个开放的、连通的子集Ω吨.
目标是恢复参数向量θ（直到刚性运动）。

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

\lambda_{\mathbf{f}}(\mathbf{x})=\sup {\lambda}\left{\lambda:|\mathbf{x}-\mathbf{f}(\lambda)|=\inf { \mu}|\mathbf{x}-\mathbf{f}(\mu)|\right}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 机器学习代写|流形学习代写manifold data learning代考|Nonlinear Manifold Learning

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习manifold data learning方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写流形学习manifold data learning代写方面经验极为丰富，各种代写流形学习manifold data learning相关的作业也就用不着说。

• 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|Nonlinear Manifold Learning

We next discuss some algorithmic techniques that proved to be innovative in the study of nonlinear manifold learning: ISOMAP, LOCAL LINEAR EMBEDDING, LAPLACIAN EIGENMAPS, DIFFUSION MAPS, HESSIAN EIGENMAPS, and the many different versions of NONLINEAR PCA. The goal of each of these algorithms is to recover the full low-dimensional

representation of an unknown nonlinear manifold, $\mathcal{M}$, embedded in some high-dimensional space, where it is important to retain the neighborhood structure of $\mathcal{M}$. When $\mathcal{M}$ is highly nonlinear, such as the S-shaped manifold in the left panel of Figure $1.1$, these algorithms outperform the usual linear techniques. The nonlinear manifold-learning methods emphasize simplicity and avoid optimization problems that could produce local minima.

Assume that we have a finite random sample of data points, $\left{\mathbf{y}{i}\right}$, from a smooth $t$ dimensional manifold $\mathcal{M}$ with metric given by the geodesic distance $d^{\mathcal{M}} ;$ see Section $1.2 .4$. These points are then nonlinearly embedded by a smooth map $\psi$ into high-dimensional input space $\mathcal{X}=\Re^{r}(t \ll r)$ with Euclidean metric $|$. $| \mathcal{X}$. This embedding provides us with the input data $\left{\mathbf{x}{i}\right}$. For example, in the right panel of Figure $1.1$, we randomly generated 20,000 three-dimensional points to lie uniformly on the surface of the two-dimensional Sshaped curve displayed in the left panel. Thus, $\psi: \mathcal{M} \rightarrow \mathcal{X}$ is the embedding map, and a point on the manifold, $\mathbf{y} \in \mathcal{M}$, can be expressed as $\mathbf{y}=\phi(\mathbf{x}), \mathbf{x} \in \mathcal{X}$, where $\phi=\psi^{-1}$. The goal is to recover $\mathcal{M}$ and find an implicit representation of the map $\psi$ (and, hence, recover the $\left.\left{\mathbf{y}{i}\right}\right)$, given only the input data points $\left{\mathbf{x}{i}\right}$ in $\mathcal{X}$.

Each algorithm computes $t^{\prime}$-dimensional estimates, $\left{\widehat{\mathbf{y}}{i}\right}$, of the $t$-dimensional manifold data, $\left{\mathbf{y}{i}\right}$, for some $t^{\prime}$. Such a reconstruction is deemed to be successful if $t^{\prime}=t$, the true (unknown) dimensionality of $\mathcal{M}$. In practice, $t^{\prime}$ will most likely be too large. Because we require a low-dimensional solution, we retain only the first two or three of the coordinate vectors and plot the corresponding elements of those vectors against each other to yield $n$ points in two- or three-dimensional space. For all practical purposes, such a display is usually sufficient to identify the underlying manifold.

Most of the nonlinear manifold-learning algorithms that we discuss here are based upon different philosophies regarding how one should recover unknown nonlinear manifolds. However, they each consist of a three-step approach (except NONLINEAR PCA). The first and third steps are common to all algorithms: the first step incorporates neighborhood information at each data point to construct a weighted graph having the data points as vertices, and the third step is a spectral embedding step that involves an $(n \times n)$-eigenequation computation. The second step is specific to the algorithm, taking the weighted neighborhood graph and transforming it into suitable input for the spectral embedding step.

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

The isometric feature mapping (or IsOMAP) algorithm (Tenenbaum, de Silva, and Langford, 2000 ) assumes that the smooth manifold $\mathcal{M}$ is a convex region of $\Re^{t}(t \ll r)$ and that the embedding $\psi: \mathcal{M} \rightarrow \mathcal{X}$ is an isometry. This assumption has two key ingredients:

• Isometry: The geodesic distance is invariant under the map $\psi$. For any pair of points on the manifold, $\mathbf{y}, \mathbf{y}^{\prime} \in \mathcal{M}$, the geodesic distance between those points equals the Euclidean distance between their corresponding coordinates, $\mathbf{x}, \mathbf{x}^{\prime} \in \mathcal{X}$; i.e.,
$$d^{\mathcal{M}}\left(\mathbf{y}, \mathbf{y}^{\prime}\right)=\left|\mathbf{x}-\mathbf{x}^{\prime}\right|_{\mathcal{X}}$$
where $\mathbf{y}=\phi(\mathbf{x})$ and $\mathbf{y}^{\prime}=\phi\left(\mathbf{x}^{\prime}\right)$.
• Convexity: The manifold $\mathcal{M}$ is a convex subset of $\Re^{t}$.
IsomaP considers $\mathcal{M}$ to be a convex region possibly distorted in any of a number of ways (e.g., by folding or twisting). The so-called Swiss roll, ${ }^{2}$ which is a flat two-dimensional rectangular submanifold of $\Re^{3}$, is one such example; see Figure $1.2$. Empirical studies show that IsOMAP works well for intrinsically flat submanifolds of $\mathcal{X}=\Re^{r}$ that look like rolledup sheets of paper or “open” manifolds such as an open box or open cylinder. However, IsOMAP does not perform well if there are any holes in the roll, because this would violate the convexity assumption. The isometry assumption appears to be reasonable for certain types of situations, but, in many other instances, the convexity assumption may be too restrictive (Donoho and Grimes, 2003b).

IsomAP uses the isometry and convexity assumptions to form a nonlinear generalization of multidimensional scaling (MDS). Recall that MDS looks for a low-dimensional subspace in which to embed input data while preserving the Euclidean interpoint distances (see Section 1.3.2). Unfortunately, working with Euclidean distances in MDS when dealing with curved regions tends to give poor results. IsOMAP follows the general MDS philosophy by attempting to preserve the global geometric properties of the underlying nonlinear manifold, and it does this by approximating all pairwise geodesic distances (i.e., lengths of the shortest paths between two points) on the manifold. In this sense, IsOMAP provides a global approach to manifold learning.

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

The Laplacian eigenmap algorithm (Belkin and Niyogi, 2002) also consists of three steps. The first and third steps of the Laplacian eigenmap algorithm are very similar to the first and third steps, respectively, of the LLE algorithm.

1. Nearest-neighbor search. Fix an integer $K$ or an $\epsilon>0$. The neighborhoods of each data point are symmetrically defined: for a $K$-neighborhood $N_{i}^{K}$ of the point $\mathbf{x}{i}$, let $\mathbf{x}{j} \in N_{i}^{K}$ iff $\mathbf{x}{i} \in N{j}^{K}$; similarly, for an $\epsilon$-neighborhood $N_{i}^{\epsilon}$, let $\mathbf{x}{j} \in N{i}^{e}$ iff $\left|\mathbf{x}{i}-\mathbf{x}{j}\right|<\epsilon$, where the norm is Euclidean norm. In general, let $N_{i}$ denote the neighborhood of $\mathbf{x}_{i}$.
2. Weighted adjacency matrix. Let $\mathbf{W}=\left(w_{i j}\right)$ be a symmetric $(n \times n)$ weighted adjacency matrix defined as follows:
$$w_{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{j}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { otherwise }\end{cases}$$

These weights are determined by the isotropic Gaussian kernel (also known as the heat kernel), with scale parameter $\sigma$. Denote the resulting weighted graph by $\mathcal{G}$. If $\mathcal{G}$ is not connected, apply step 3 to each connected subgraph.

1. Spectral embedding. Let $\mathbf{D}=\left(d_{i j}\right)$ be an $(n \times n)$ diagonal matrix with diagonal elements $d_{i i}=\sum_{j \in N_{i}} w_{i j}=\left(\mathbf{W} 1_{n}\right){i}, i=1,2, \ldots, n$. The $(n \times n)$ symmetric matrix $\mathbf{L}=\mathbf{D}-\mathbf{W}$ is known as the graph Laplacian for the graph $\mathcal{G}$. Let $\mathbf{y}=\left(y{i}\right)$ be an $n$-vector. Then, $\mathbf{y}^{\tau} \mathbf{L} \mathbf{y}=\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} w_{i j}\left(y_{i}-y_{j}\right)^{2}$, so that $\mathbf{L}$ is nonnegative definite.

When data are uniformly sampled from a low-dimensional manifold $\mathcal{M}$ of $\Re^{r}$, the graph Laplacian $\mathbf{L}=\mathbf{L}{n, \sigma}$ (considered as a function of $n$ and $\sigma$ ) can be regarded as a discrete approximation to the continuous Laplace-Beltrami operator $\Delta{\mathcal{M}}$ defined on the manifold $\mathcal{M}$, and converges to $\Delta_{\mathcal{M}}$ as $\sigma \rightarrow 0$ and $n \rightarrow \infty$. Furthermore, when the data are sampled from an arbitrary probability distribution $P$ on the manifold $\mathcal{M}$, then, under certain conditions on $\mathcal{M}$ and $P$, the graph Laplacian converges to a weighted version of $\Delta_{\mathcal{M}}$ (Belkin and Niyogi, 2008).

The $(t \times n)$-matrix $\mathbf{Y}=\left(\mathbf{y}{1}, \cdots, \mathbf{y}{n}\right)$, which is used to embed the graph $\mathcal{G}$ into the low-dimensional space $\Re^{t}$, where $\mathbf{y}{i}$ yields the embedding coordinates of the $i$ th point, is determined by minimizing the objective function, $$\sum{i} \sum_{j} w_{i j}\left|\mathbf{y}{i}-\mathbf{y}{j}\right|^{2}=\operatorname{tr}\left{\mathbf{Y L Y} \mathbf{Y}^{\tau}\right}$$
In other words, we seek the solution,
$$\widehat{\mathbf{Y}}=\arg \min {\mathbf{Y}: \mathbf{Y D \mathbf { Y } ^ { \top } = \mathbf { I } { t }}} \operatorname{tr}\left{\mathbf{Y L Y ^ { \top } }},\right.$$
where we restrict $\mathbf{Y}$ such that $\mathbf{Y D Y} \mathbf{Y}^{\tau}=\mathbf{I}{t}$ to prevent a collapse onto a subspace of fewer than $t-1$ dimensions. The solution is given by the generalized eigenequation, $\mathbf{L v}=\lambda \mathbf{D v}$, or, equivalently, by finding the eigenvalues and eigenvectors of the matrix $\widehat{\mathbf{W}}=\mathbf{D}^{-1 / 2} \mathbf{W D} \mathbf{D}^{-1 / 2}$. The smallest eigenvalue, $\lambda{n}$, of $\widehat{\mathbf{W}}$ is zero. If we ignore the smallest eigenvalue (and its corresponding constant eigenvector $\mathbf{v}{n}=1{n}$ ), then the best embedding solution in $\Re^{t}$ is similar to that given by LLE; that is, the rows of $\widehat{\mathbf{Y}}$ are the eigenvectors,
$$\widehat{\mathbf{Y}}=\left(\widehat{\mathbf{y}}{1}, \cdots, \widehat{\mathbf{y}}{n}\right)=\left(\mathbf{v}{n-1}, \cdots, \mathbf{v}{n-t}\right)^{\tau},$$
corresponding to the next $t$ smallest eigenvalues, $\lambda_{n-1} \leq \cdots \leq \lambda_{n-t}$, of $\widehat{\mathbf{W}}$.

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

• 等距：测地线距离在地图下是不变的ψ. 对于流形上的任意一对点，是,是′∈米，这些点之间的测地线距离等于它们对应坐标之间的欧几里得距离，X,X′∈X; IE，
d米(是,是′)=|X−X′|X
在哪里是=φ(X)和是′=φ(X′).
• 凸性：流形米是一个凸子集ℜ吨.
IsomaP 认为米成为可能以多种方式（例如，通过折叠或扭曲）中的任何一种方式扭曲的凸面区域。所谓的瑞士卷，2它是一个平面二维矩形子流形ℜ3, 就是这样一个例子；见图1.2. 实证研究表明，IsOMAP 适用于本质平坦的子流形X=ℜr看起来像卷起的纸或“打开”的歧管，例如打开的盒子或打开的圆柱体。但是，如果卷中有任何孔洞，IsOMAP 的性能就不会很好，因为这会违反凸性假设。对于某些类型的情况，等距假设似乎是合理的，但在许多其他情况下，凸性假设可能过于严格（Donoho 和 Grimes，2003b）。

IsomAP 使用等距和凸性假设来形成多维尺度 (MDS) 的非线性泛化。回想一下，MDS 寻找一个低维子空间，在其中嵌入输入数据，同时保留欧几里德点间距离（参见第 1.3.2 节）。不幸的是，在处理弯曲区域时，在 MDS 中使用欧几里德距离往往会产生较差的结果。IsOMAP 遵循一般 MDS 哲学，试图保留底层非线性流形的全局几何特性，它通过近似流形上的所有成对测地线距离（即两点之间的最短路径的长度）来做到这一点。从这个意义上说，IsOMAP 为流形学习提供了一种全局方法。

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

1. 最近邻搜索。修复一个整数ķ或一个ε>0. 每个数据点的邻域是对称定义的：对于ķ-邻里ñ一世ķ点的X一世， 让Xj∈ñ一世ķ当且当X一世∈ñjķ; 同样，对于一个ε-邻里ñ一世ε， 让Xj∈ñ一世和当且当|X一世−Xj|<ε，其中范数是欧几里得范数。一般来说，让ñ一世表示邻域X一世.
2. 加权邻接矩阵。让在=(在一世j)是对称的(n×n)加权邻接矩阵定义如下：
w_{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{ x}{i}-\mathbf{x}{j}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \在 N{i} \ 0, & \text { 否则 }\end{cases}w_{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{ x}{i}-\mathbf{x}{j}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \在 N{i} \ 0, & \text { 否则 }\end{cases}

1. 光谱嵌入。让D=(d一世j)豆(n×n)具有对角元素的对角矩阵d一世一世=∑j∈ñ一世在一世j=(在1n)一世,一世=1,2,…,n. 这(n×n)对称矩阵大号=D−在被称为图 Laplacian for the graphG. 让是=(是一世)豆n-向量。然后，是τ大号是=12∑一世=1n∑j=1n在一世j(是一世−是j)2， 以便大号是非负定的。

\widehat{\mathbf{Y}}=\arg \min {\mathbf{Y}: \mathbf{Y D \mathbf { Y } ^ { \top } = \mathbf { I } { t }}} \operatorname{tr }\left{\mathbf{Y L Y ^ { \top } }},\right.\widehat{\mathbf{Y}}=\arg \min {\mathbf{Y}: \mathbf{Y D \mathbf { Y } ^ { \top } = \mathbf { I } { t }}} \operatorname{tr }\left{\mathbf{Y L Y ^ { \top } }},\right.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。