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

## 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。