# 机器学习代写|流形学习代写manifold data learning代考|ACDL 2022

#### Doug I. Jones

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couryes-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
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## 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.

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

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.

# 流形学习代写

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

$$\hat{f} m(\mathbf{y} 0)=\frac{1}{m} \sum_{i=1}^{m} \frac{1}{h_{m}^{D}} K\left(\frac{|\mathbf{y} i-\mathbf{y} 0|}{h_{m}}\right)$$

$$\operatorname{MSE}[\hat{f} m(\mathbf{y} 0)]=\mathrm{E}\left[(\hat{f} m(\mathbf{y} 0)-f(\mathbf{y} 0))^{2}\right]=O\left(h_{m}^{4}+\frac{1}{m h_{m}^{D}}\right)$$

$$\operatorname{MSE}[\hat{f} m(p)]=O\left(\frac{1}{m^{4 /(D+4)}}\right)$$

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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。

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