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

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 Manifolds

It was not until 1936 that the first clear description of the nature of an abstract manifold was provided by Hassler Whitney. We regard a “manifold” as a generalization to higher dimensions of a curved surface in three dimensions. A topological space $\mathcal{M}$ is a topological manifold of dimension $d$ (sometimes written as $\mathcal{M}^{d}$ ) if it is a second-countable Hausdorff space that is also locally Euclidean of dimension $d$. The last condition says that at every point on the manifold, there exists a small local region such that the manifold enjoys the properties of Euclidean space. The Hausdorff condition ensures that distinct points on the manifold can be separated (by neighborhoods), and the second-countability condition ensures that the manifold is not too large. The two conditions of Hausdorff and second countability, together with an embedding theorem of Whitney (1936), imply that any $d$ dimensional manifold, $\mathcal{M}^{d}$, can be embedded in $\mathfrak{R}^{2 d+1}$. In other words, a space of at most $2 d+1$ dimensions is required to embed a $d$-dimensional manifold. A submanifold is just a manifold lying inside another manifold of higher dimension. As a topological space, a manifold can have topological structure, such as being compact or connected.

## 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 Habrlitations 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 $\mathbf{y} \in V, g(\mathbf{x})=$ $\mathbf{y}$. 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 $\mathcal{C}^{\text {T}}$-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.

Consider a point $\mathrm{p} \in \mathcal{M}$. The set, $T_{\mathbf{p}}(\mathcal{M})$, of all vectors that are tangent to the manifold at the point $p$ forms a vector space called the tangent space at p. The tangent space has the same dimension as $\mathcal{M}$. Each tangent space $T_{\mathbf{p}}(\mathcal{M})$ at a point $\mathbf{p}$ has an inner-product, $g_{\mathbf{p}}=\langle\cdot, \cdot): T_{\mathbf{p}}(\mathcal{M}) \times T_{\mathbf{p}}(\mathcal{M}) \rightarrow \Re$, which is defined to vary smoothly over the manifold with $\mathbf{p}$. For $\mathbf{x}, \mathbf{y}, \mathbf{z} \in T_{\mathbf{p}}(\mathcal{M})$, the inner-product $g_{\mathbf{p}}$ is
bilinear: $g_{\mathbf{p}}(a \mathbf{x}+b \mathbf{y}, \mathbf{z})=a g_{\mathbf{p}}(\mathbf{x}, \mathbf{z})+b g_{\mathbf{p}}(\mathbf{y}, \mathbf{z})$, for $a, b \in \Re$,

symmetric: $g_{\mathbf{p}}(\mathbf{x}, \mathbf{y})=g_{\mathbf{p}}(\mathbf{y}, \mathbf{x})$,
positive-definite: $g_{\mathbf{p}}(\mathbf{x}, \mathbf{y}) \geq 0$ and $g_{\mathbf{p}}(\mathbf{x}, \mathbf{x})=0$ iff $\mathbf{x}=\mathbf{0}$.
The collection of inner-products $g=\left{g_{\mathbf{p}}: \mathbf{p} \in \mathcal{M}\right}$ is a Riemannian metric on $\mathcal{M}$, and the pair $(\mathcal{M}, g)$ defines a Riemannian manifold.

Suppose $\left(\mathcal{M}, g^{\mathcal{M}}\right)$ and $\left(\mathcal{N}, g^{\mathcal{N}}\right)$ are two Riemannian manifolds that have the same dimension, and let $\psi: \mathcal{M} \rightarrow \mathcal{N}$ be a diffeomorphism. Then, $\psi$ is an isometry if for all $\mathbf{p} \in \mathcal{M}$ and any two points $\mathbf{u}, \mathbf{v} \in T_{\mathbf{p}}(\mathcal{M}), g^{\mathcal{M}}(\mathbf{u}, \mathbf{v})=g^{\mathcal{N}}(\psi(\mathbf{u}), \psi(\mathbf{v}))$; in other words, $\psi$ is an isometry if $\psi$ “pulls back” one Riemannian metric to the other.

## 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 $c(\lambda)=\left(c_{1}(\lambda), \cdots, c_{d}(\lambda)\right)^{\tau}$ 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
$$L(c)=\int_{\lambda_{0}}^{\lambda_{1}}\left|c^{\prime}(\lambda)\right| d \lambda$$
In the event that a curve has unit speed, its arc-length is $L(c)=\lambda_{1}-\lambda_{0}$.
Example: The Unit Circle in $\Re^{2}$. The unit circle in $\Re^{2}$, which is defined as $\left{\left(x_{1}, x_{2}\right) \in \Re^{2}\right.$ : $\left.x_{1}^{2}+x_{2}^{2}=1\right}$, is a one-dimensional curve that can be parametrized as
$$c(\lambda)=\left(c_{1}(\lambda), c_{2}(\lambda)\right)^{\tau}=(\cos \lambda, \sin \lambda)^{\top}, \quad \lambda \in[0,2 \pi)$$
The unit circle is a closed curve, its velocity is $c^{\prime}(\lambda)=(-\sin \lambda, \cos \lambda)^{\tau}$, and its speed is $\left|c^{\prime}(\lambda)\right|=1$.

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

C′(λ)=(C1′(λ),⋯,Cd′(λ))τ,

\left|c^{\prime}(\lambda)\right|=\left{\sum_{j=1}^{d}\left[c_{j}^{\prime}(\lambda)\right] ^{2}\right}^{1 / 2} 。\left|c^{\prime}(\lambda)\right|=\left{\sum_{j=1}^{d}\left[c_{j}^{\prime}(\lambda)\right] ^{2}\right}^{1 / 2} 。

C(λ)=(C1(λ),C2(λ))τ=(因⁡λ,罪⁡λ)⊤,λ∈[0,2圆周率)

## 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。