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

#### Doug I. Jones

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

Much of the recent work in manifold learning and nonlinear dimensionality reduction focuses on distance-based methods, i.e., methods that aim to preserve the local or global (geodesic) distances between data points on a submanifold of Euclidean space. While this is a promising approach when the data manifold is known to have no intrinsic curvature (which is the case for common examples such as the “Swiss roll”), classical results in Riemannian geometry show that it is impossible to map a $d$-dimensional data manifold with intrinsic curvature into $\mathbb{R}^{d}$ in a manner that preserves distances. Consequently, distance-based methods of dimensionality reduction distort intrinsically curved data spaces, and they often do so in unpredictable ways. In this chapter, we discuss an alternative paradigm of manifold learning. We show that it is possible to perform nonlinear dimensionality reduction by preserving the underlying density of the data, for a much larger class of data manifolds than intrinsically flat ones, and demonstrate a proof-of-concept algorithm demonstrating the promise of this approach.

Visual inspection of data after dimensional reduction to two or three dimensions is among the most common uses of manifold learning and nonlinear dimensionality reduction. Typically, what is sought by the user’s eye in two or three-dimensional plots is clustering and other relationships in the data. Knowledge of the density, in principle, allows one to identify such basic structures as clusters and outliers, and even define nonparametric classifiers; the underlying density of a data set is arguably one of the most fundamental statistical objects that describe it. Thus, a method of dimensionality reduction that is guaranteed to preserve densities may well be preferable to methods that aim to preserve distances, but end up distorting them in uncontrolled ways.

Many of the manifold learning methods require the user to set a neighborhood radius $h$, or, for $k$-nearest neighbor approaches, a positive integer $k$, to be used in determining the neighborhood graph. Most of the time, there is no automatic way to pick the appropriate values of the tweak parameters $h$ and $k$, and one resorts to trial and error, looking for values that result in reasonable-looking plots. Kernel density estimation, one of the most popular and useful methods of estimating the underlying density of a data set, comes with a natural way to choose $h$ or $k$; it suggests to us to pick the value that maximizes a cross-validation score for the density estimate. While the usual kernel density estimation does not allow one to estimate the density of data on submanifolds of Euclidean space, a small modification allows one to do so. This modification and its ramifications are discussed below in the context of density-preserving maps.

## 机器学习代写|流形学习代写manifold data learning代考|Dimensional Reduction to R

These results were formulated in terms of so-called closed manifolds, i.e., compact manifolds without boundary. The practical dimensionality reduction problem we would like to address, on the other hand, involves starting with a $d$-dimensional data submanifold $M$ of $\mathbb{R}^{D}$ (where $d<D$ ), and dimensionally reducing to $\mathbb{R}^{d}$. In order to be able to do this diffeomorphically, $M$ must be diffeomorphic to a subspace of $\mathbb{R}^{d}$, which is not generally the case for closed manifolds. For instance, although we can find a diffeomorphism from a hemisphere (a manifold with boundary, not a closed manifold) into the plane, we cannot find one from the unit sphere (a closed manifold) into the plane. This is a constraint on all dimensional reduction algorithms that preserve the global topology of the data space, not just density preserving maps. Any algorithm that aims to avoid “tearing” or “folding” the data subspace during the reduction will fail on problems like reducing a sphere to $\mathbb{R}^{2} .5$

Thus, in order to show that density preserving maps into $\mathbb{R}^{d}$ exist for a useful class of $d$-dimensional data manifolds, we have to make sure that the conclusion of Moser’s theorem and our corollary work for certain manifolds with boundary, or for certain non-compact manifolds, as well. Fortunately, this is not so hard, at least for a simple class of manifolds that is enough to be useful. In proving his theorem for closed manifolds, Moser [18] first gives a proof for a single “coordinate patch” in such a manifold, which, basically, defines a compact manifold with boundary minus the boundary itself. Not all $d$-dimensional manifolds with boundary (minus their boundaries) can be given by atlases consisting of a single coordinate patch, but the ones that can be so given cover a wide range of curved Riemannian manifolds, including the hemisphere and the Swiss roll, possibly with punctures. In the following, we will assume that $M$ consists of a single coordinate patch.

# 流形学习代写

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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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