### 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
$$c^{\prime}(\lambda)=\left(c_{1}^{\prime}(\lambda), \cdots, c_{d}^{\prime}(\lambda)\right)^{\tau}$$

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

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## MATLAB代写

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