### 机器学习代写|流形学习代写manifold data learning代考|Preserving the Estimated Density

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|The Optimization

Now that we have a method to estimate the density on a submanifold of $\mathbb{R}^{D}$, we can proceed to define an algorithm for density preserving maps. ${ }^{9}$ Suppose we are given a sample $X=\left{x_{1}, x_{2}, \ldots, x_{m}\right}$ of $m$ data points $x_{i} \in \mathbb{R}^{D}$ that live on a $d$-dimensional submanifold $M$ of $\mathbb{R}^{D}$. We first proceed to estimate the density at each one of the points, by using a slightly generalized version of the submanifold estimator that has variable bandwidths. Denoting the bandwidth for a given evaluation point $x_{j}$ and a reference (data) point $x_{i}$ by $h_{i j}$, the generalized, variable bandwidth estimator at $x_{j}$ is, ${ }^{10}$
$$\hat{f_{j}}=\hat{f}\left(x_{j}\right)=\frac{1}{m} \sum_{i} \frac{1}{h_{i j}^{d}} K_{d}\left(\frac{\left|x_{j}-x_{i}\right|_{D}}{h_{i j}}\right) .$$
Variable bandwidth methods allow the estimator to adapt to the inhomogeneities in the data. Various approaches exist for picking the bandwidths $h_{i j}$ as functions of the query (evaluation) point $x_{j}$ and/or the reference point $x_{i}[25]$. Here, we focus on the $k$ th-nearest neighbor approach for evaluation points, i.e., we take $h_{i j}$ to depend only on the evaluation point $x_{j}$, and we let $h_{i j}=h_{j}=$ the distance of the $k$ th nearest data (reference) point to the evaluation point $x_{j}$. Here, $k$ is a free parameter that needs to be picked by the user. However, instead of tuning it by hand, one can use a leave-one-out cross-validation score [25] such as the log-likelihood score for the density estimate to pick the best value. This is done by estimating the log-likelihood of each data point by using the leave-one-out version

of the density estimate $(3.7)$ for a range of $k$ values, and picking the $k$ that gives the highest log-likelihood.

Now, given the estimates $\hat{f}{j}=\hat{f}\left(x{j}\right)$ of the submanifold density at the $D$-dimensional data points $x_{j}$, we want to find a $d$-dimensional representation $X^{\prime}=\left{x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{m}^{\prime}\right}$, $x_{i}^{\prime} \in \mathbb{R}^{d}$ such that the new estimates $\hat{f}{i}^{\prime}$ at the points $x{i}^{\prime} \in \mathbb{R}^{d}$ agree with the original density estimates, i.e.,
$$\hat{f}{i}^{\prime}=\hat{f}{i}, \quad i=1, \ldots, m .$$
For this purpose, one can attempt, for example, to minimize the mean squared deviation of $\hat{f}{i}^{\prime}$ from $\hat{f}{i}$ as a function of the $x_{i}^{\prime}$ s, but such an approach would result in a non-convex optimization problem with many local minima. We formulate an alternative approach involving semidefinite programming, for the special case of the Epanechnikov kernel [25], which is known to be asymptotically optimal for density estimation, and is convenient for formulating a convex optimization problem for the matrix of inner products (the Gram matrix, or the kernel matrix) of the low dimensional data set $X^{\prime}$.

## 机器学习代写|流形学习代写manifold data learning代考|The Optimization

The Epanechnikov kernel. The Epanechnikov kernel $k_{e}$ in $d$ dimensions is defined as,
$$k_{e}\left(\left|x_{i}-x_{j}\right|\right)=\left{\begin{array}{cc} N_{e}\left(1-\left|x_{i}-x_{j}\right|^{2}\right), & 0 \leq\left|x_{i}-x_{j}\right| \leq 1 \ 0, & 1 \leq\left|x_{i}-x_{j}\right| \end{array}\right.$$
where $N_{e}$ is the normalization constant that ensures $\int_{\mathbb{R}{d}} k{e}\left(\left|x-x^{\prime}\right|\right) d^{d} x^{\prime}=1$. We will assume that the kernel used in the estimates $\hat{f}{i}$ and $\hat{f}{i}^{\prime}$ of the density via (3.7) is the Epanechnikov kernel. Owing to its quadratic form (3.9), this kernel facilitates the formulation of a convex optimization problem. Instead of seeking the dimensionally reduced version $X^{\prime}=\left{x_{1}^{\prime}, \ldots, x_{n}^{\prime}\right}$ of the data set directly, we will first aim to obtain the kernel matrix $K_{i j}=x_{i}^{\prime} \cdot x_{j}^{\prime}$ for the low-dimensional data points. This is a common approach in the manifold learning literature, where one obtains the low-dimensional data points themselves from the $K_{i j}$ via a singular value decomposition.

We next formulate the DPM optimization problem using the Epanechnikov kernel, and comment on the motivation behind it. As in the case of distance-based manifold learning methods, there will likely be various approaches to density-preserving dimensional reduction, some computationally more efficient than the one discussed here. We hope the discussions in this chapter will stimulate further research in this area.

Given the estimated densities $\hat{f}{i}$, we seek a symmetric, positive semidefinite inner product matrix $K{i j}=x_{i}^{\prime} \cdot x_{j}^{\prime}$ that results in $d$-dimensional density estimates that agree with $\hat{f}_{i}$. In order to deal with the non-uniqueness problem mentioned during our discussion of densitypreserving maps between manifolds (which likely carries over to the discrete setting), we need to pick a suitable objective function to maximize. We choose the objective function to be the same as that of Maximum Variance Unfolding (MVU) [29], namely, $\operatorname{trace}(K)$. After getting rid of translations by constraining the center of mass of the dimensionally reduced data points to the origin, maximizing the objective function trace $(K)$ becomes equivalent to maximizing the sum of the squared distances between the data points [29].

While the objective function for DPM is the same as that of MVU, the constraints of the former will be weaker. Instead of preserving the distances between $k$-nearest neighbors, the DPM optimization defined below preserves the total contribution of the original $k$-nearest neighbors to the density estimate at the data points. As opposed to MVU, this allows for local stretches of the data set, and results in optimal kernel matrices $K$ that can be faithfully represented by a smaller number of dimensions than the intrinsic dimensionality suggested by MVU. For instance, while MVU is capable of unrolling data on the Swiss roll onto a flat plane, it is impossible to lay data from a spherical cap onto the plane while keeping the distances to the $k$ th nearest neighbors fixed. ${ }^{11}$ Thus, the constraints of the optimization in MVU are too stringent to give an inner product matrix $K$ of rank 2, when the original data is on an intrinsically curved surface in $\mathbb{R}^{3}$. We will see below that the looser constraints of DPM allow it to do a better job in capturing the intrinsic dimensionality of a curved surface.

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

In this chapter, we discussed density preserving maps, a density-based alternative to distancebased methods of manifold learning. This method aims to perform dimensionality reduction on large-dimensional data sets in a way that preservs their density. By using a classical result due to Moser, we proved that density preserving maps to $\mathbb{R}^{d}$ exist even for data on intrinsically curved $d$-dimensional submanifolds of $\mathbb{R}^{D}$ that are globally, or topologically “simple.” Since the underlying probability density function is arguably one of the most fundamental statistical quantities pertaining to a data set, a method that preserves densities while performing dimensionality reduction is guaranteed to preserve much valuable structure in the data. While distance-preserving approaches distort data on intrinsically curved spaces in various ways, density preserving maps guarantee that certain fundamental statistical information is conserved.

We reviewed a method of estimating the density on a submanifold of Euclidean space. This method was a slightly modified version of the classical method of kernel density estimation, with the additional property that the convergence rate was determined by the intrinsic dimensionality of the data, instead of the full dimensionality of the Euclidean space the data was embedded in. We made a further modification on this estimator to allow for variable “bandwidths,” and used it with a specific kernel function to set up a semidefinite optimization problem for a proof-of-concept approach to density preserving maps. The objective function used was identical to the one in Maximum Variance Unfolding [29], but the constraints were significantly weaker than the distance-preserving constraints in MVU. By testing the methods on two relatively small, synthetic data sets, we experimentally confirmed the theoretical expectations and showed that density preserving maps are better in detecting and reducing to the intrinsic dimensionality of the data than some of the commonly used distance-based approaches that also work by first estimating a kernel matrix.
While the initial formulation presented in this chapter is not yet scalable to large data sets, we hope our discussion will motivate our readers to pursue the idea of density preserving maps further, and explore alternative, superior formulations. One possible approach to speeding up the computation is to use fast semidefinite programming techniques [4].

## 机器学习代写|流形学习代写manifold data learning代考|The Optimization

Fj^=F^(Xj)=1米∑一世1H一世jdķd(|Xj−X一世|DH一世j).

F^一世′=F^一世,一世=1,…,米.

## 机器学习代写|流形学习代写manifold data learning代考|The Optimization

Epanechnikov 内核。Epanechnikov 内核ķ和在d维度定义为
$$k_{e}\left(\left|x_{i}-x_{j}\right|\right)=\left{ñ和(1−|X一世−Xj|2),0≤|X一世−Xj|≤1 0,1≤|X一世−Xj|\对。$$

## 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。