## 计算机代写|流形学习代写Manifold learning代考|Math214

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习Manifold 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 learning代写方面经验极为丰富，各种代写流形学习Manifold learning相关的作业也就用不着说。

## 计算机代写|流形学习代写Manifold 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 $N_i^\epsilon$ of the point $\mathbf{x}_i$ as in Step 1 of Laplacian eigenmaps. In general, let $N_i$ denote the neighborhood of $\mathbf{x}_i$.
1. 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, $$w{i j}=w\left(\mathbf{x}i, \mathbf{x}_j\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{x}_i-\mathbf{x}_j\right|^2}{2 \sigma^2}\right}, & \text { if } \mathbf{x}_j \in N_i ; \ 0, & \text { otherwise. }\end{cases}$$ 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 $\mathbf{W}$ gives information on the local geometry of the data.
2. Spectral embedding. Define $\mathbf{D}=\left(d_{i j}\right)$ to be a diagonal matrix formed from the matrix $\mathbf{W}$ 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
$$\mathbf{P}=\mathbf{D}^{-1 / 2} \mathbf{L} \mathbf{D}^{-1 / 2}=\mathbf{I}_n-\mathbf{D}^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2},$$
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 $\mathcal{G}$.

## 计算机代写|流形学习代写Manifold 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 $\boldsymbol{\theta} \in \Theta$. 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 $\phi: \Theta \rightarrow \Re^r$, 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 $\mathrm{x}$ 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,
$$d^{\mathcal{M}}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|\boldsymbol{\theta}-\boldsymbol{\theta}^{\prime}\right|_{\Theta},$$
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 $\Re^t$.

# 流形学习代写

## 计算机代写|流形学习代写Manifold learning代考|Diffusion Maps

$$\mathbf{P}=\mathbf{D}^{-1 / 2} \mathbf{L} \mathbf{D}^{-1 / 2}=\mathbf{I}_n-\mathbf{D}^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2},$$

## 计算机代写|流形学习代写Manifold learning代考|Hessian Eigenmaps

Hessian Eigenmaps (Donoho and Grimes, 2003b)被提出用于恢复经常违反凹凸性假设的高维铰接图像库的流形。设$\Theta \subset \Re^t$为参数空间，假设$\phi: \Theta \rightarrow \Re^r$，其中$t<r$。假设$\mathcal{M}=\phi(\Theta)$是一个平滑的铰接图像集合。IsOMAP的等距和凹凸性要求被以下较弱的要求所取代:

$$d^{\mathcal{M}}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|\boldsymbol{\theta}-\boldsymbol{\theta}^{\prime}\right|_{\Theta},$$

## 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。

## 计算机代写|流形学习代写Manifold learning代考|Math214

statistics-lab™ 为您的留学生涯保驾护航 在代写流形学习Manifold 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 learning代写方面经验极为丰富，各种代写流形学习Manifold learning相关的作业也就用不着说。

## 计算机代写|流形学习代写Manifold 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 $|\cdot|{\mathcal{X}}$. 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.

## 计算机代写|流形学习代写Manifold 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.

# 流形学习代写

## 计算机代写|流形学习代写Manifold learning代考|Isomap

$$d^{\mathcal{M}}\left(\mathbf{y}, \mathbf{y}^{\prime}\right)=\left|\mathbf{x}-\mathbf{x}^{\prime}\right|_{\mathcal{X}}$$

IsomaP认为$\mathcal{M}$是一个凸区域，可能以多种方式(例如，折叠或扭曲)中的任何一种方式扭曲。所谓的瑞士卷${ }^2$是$\Re^3$的平面二维矩形子流形，就是这样一个例子;见图1.2。实证研究表明，IsOMAP可以很好地用于$\mathcal{X}=\Re^r$的本质上扁平的子流形，这些子流形看起来像卷起的纸张或“开放”的流形，如开放的盒子或开放的圆柱体。然而，如果卷中有孔，IsOMAP就不能很好地执行，因为这会违反凹凸性假设。在某些情况下，等长假设似乎是合理的，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来嵌入输入数据，同时保持欧几里得点间距离(参见第1.3.2节)。不幸的是，当处理弯曲区域时，在MDS中使用欧几里得距离往往会得到很差的结果。IsOMAP遵循一般的MDS原理，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两两测地线距离(即两点之间最短路径的长度)来实现这一点。从这个意义上说，IsOMAP为流形学习提供了一种全局方法。

## 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。

## 计算机代写|流形学习代写Manifold learning代考|Math214

statistics-lab™ 为您的留学生涯保驾护航 在代写流行学习Manifold 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行学习Manifold learning代写方面经验极为丰富，各种代写流行学习Manifold learning相关的作业也就用不着说。

## 计算机代写|流形学习代写Manifold 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 $\mathcal{C}^r$-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.

## 计算机代写|流形学习代写Manifold 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 $c(\lambda)=\left(c_1(\lambda), \cdots, c_d(\lambda)\right)^\tau$ 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
$$L(c)=\int_{\lambda_0}^{\lambda_1}\left|c^{\prime}(\lambda)\right| d \lambda$$

# 流形学习代写

## 计算机代写|流形学习代写Manifold learning代考|Curves and Geodesics

$\mathcal{M}$中的曲线被定义为从$\Re$中的开放区间$\Lambda$(可能有无限长)到$\mathcal{M}$的平滑映射。点$\lambda \in \Lambda$形成曲线的参数化。设$c(\lambda)=\left(c_1(\lambda), \cdots, c_d(\lambda)\right)^\tau$为$\Re^d$中的曲线，由$\lambda \in \Lambda \subseteq \Re$参数化。如果我们使$c(\lambda)$的坐标函数$\left{c_h(\lambda)\right}$尽可能光滑(通常是$\mathcal{C}^{\infty}$，具有任意数量的连续导数的函数)，那么我们说$c$是一条光滑曲线。如果$c(\lambda+\alpha)=c(\lambda)$对于所有的$\lambda, \lambda+\alpha \in \Lambda$，曲线$c$被认为是闭合的。速度(或切线)向量在点$\lambda$是由
$$c^{\prime}(\lambda)=\left(c_1^{\prime}(\lambda), \cdots, c_d^{\prime}(\lambda)\right)^\tau,$$
$c_j^{\prime}(\lambda)=d c_j(\lambda) / d \lambda$和曲线的“速度”在哪里
$$\left|c^{\prime}(\lambda)\right|=\left{\sum_{j=1}^d\left[c_j^{\prime}(\lambda)\right]^2\right}^{1 / 2} .$$


L(c)=\int_{\lambda_0}^{\lambda_1}\left|c^{\prime}(\lambda)\right| d \lambda

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。