机器学习代写|流形学习代写manifold data learning代考|Spectral Embedding Methods for Manifold Learning

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我们提供的流形学习manifold data learning及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (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|Spectral Embedding Methods for Manifold Learning

机器学习代写|流形学习代写manifold data learning代考|Alan Julian Izenman

Manifold learning encompasses much of the disciplines of geometry, computation, and statistics, and has become an important research topic in data mining and statistical learning. The simplest description of manifold learning is that it is a class of algorithms for recovering a low-dimensional manifold embedded in a high-dimensional ambient space. Major breakthroughs on methods for recovering low-dimensional nonlinear embeddings of highdimensional data (Tenenbaum, de Silva, and Langford, 2000; Roweis and Saul, 2000) led to the construction of a number of other algorithms for carrying out nonlinear manifold learning and its close relative, nonlinear dimensionality reduction. The primary tool of all embedding algorithms is the set of eigenvectors associated with the top few or bottom few eigenvalues of an appropriate random matrix. We refer to these algorithms as spectral embedding methods. Spectral embedding methods are designed to recover linear or nonlinear manifolds, usually in high-dimensional spaces.

Linear methods, which have long been considered part-and-parcel of the statistician’s toolbox, include PRINCIPAL COMPONENT ANALYSIS (PCA) and MULTIDIMENSIONAL SCALING (MDS). PCA has been used successfully in many different disciplines and applications. In computer vision, for example, PCA is used to study abstract notions of shape, appearance, and motion to help solve problems in facial and object recognition, surveillance, person tracking, security, and image compression where data are of high dimensionality (Turk and Pentland, 1991; De la Torre and Black, 2001). In astronomy, where very large digital sky surveys have become the norm, PCA has been used to analyze and classify stellar spectra, carry out morphological and spectral classification of galaxies and quasars, and analyze images of supernova remnants (Steiner, Menezes, Ricci, and Oliveira, 2009). In bioinformatics, PCA has been used to study high-dimensional data generated by genome-wide, gene-expression experiments on a variety of tissue sources, where scatterplots of the top principal components in such studies often show specific classes of genes that are expressed by different clusters of distinctive biological characteristics (Yeung and Ruzzo, 2001; ZhengBradley, Rung, Parkinson, and Brazma, 2010). PCA has also been used to select an optimal subset of single nucleotide polymorphisms (SNPs) (Lin and Altman, 2004). PCA is also

used to derive approximations to more complicated nonlinear subspaces, including problems involving data interpolation, compression, denoising, and visualization.

MDS, which has its origins in psychology, has recently been found most useful in bioinformatics, where it is known as “distance geometry.” MDS, for example, has been used to display a global representation (i.e., a map) of the protein-structure universe (Holm and Sander, 1996; Hou, Sims, Zhang, and Kim, 2003; Hou, Jun, Zhang, and Kim, 2005; Lu, Keles, Wright, and Wahba, 2005; Kim, Ahn, Lee, Park, and Kim, 2010). The idea is that points that are closely positioned to other points provide important information on the shape and function of proteins within the same family and so can be used for prediction and classification purposes. See Izenman (2008, Table 13.1) for a list of many diverse application areas and research topics in MDS.

机器学习代写|流形学习代写manifold data learning代考|Spaces and Manifolds

Manifold learning involves concepts from general topology and differential geometry. Good introductions to topological spaces include Kelley (1955), Willard (1970), Bourbaki (1989), Mendelson (1990), Steen (1995), James (1999), and several of these have since been reprinted. Books on differential geometry include Spivak (1965), Kreyszig (1991), Kühnel (2000), Lee (2002), and Pressley (2010).

Manifolds generalize the notions of curves and surfaces in two and three dimensions to higher dimensions. Before we give a formal description of a manifold, it will be helpful to visualize the notion of a manifold. Imagine an ant at a picnic, where there are all sorts of items from cups to doughnuts. The ant crawls all over the picnic items, but because of its tiny size, the ant sees everything on a very small scale as flat and featureless. Similarly, a human, looking around at the immediate vicinity, would not see the curvature of the earth. A manifold (also referred to as a topological manifold) can be thought of in similar terms, as a topological space that locally looks flat and featureless and behaves like Euclidean space. Unlike a metric space, a topological space has no concept of distance. In this Section, we review specific definitions and ideas from topology and differential geometry that enable us to provide a useful definition of a manifold.

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

Topological spaces were introduced by Maurice Fréchet (1906) (in the form of metric spaces), and the idea was developed and extended over the next few decades. Amongst those who contributed significantly to the subject was Felix Hausdorff, who in 1914 coined the phrase “topological space” using Johann Benedict Listing’s German word Topologie introduced in $1847 .$

A topological space $\mathcal{X}$ is a nonempty collection of subsets of $\mathcal{X}$ which contains the empty set, the space itself, and arbitrary unions and finite intersections of those sets. A topological space is often denoted by $(\mathcal{X}, \mathcal{T})$, where $\mathcal{T}$ represents the topology associated with $\mathcal{X}$. The elements of $\mathcal{T}$ are called the open sets of $\mathcal{X}$, and a set is closed if its complement is open. Topological spaces can also be characterized through the concept of neighborhood. If $\mathbf{x}$ is a point in a topological space $\mathcal{X}$, its neighborhood is a set that contains an open set that

contains $x$.
Let $\mathcal{X}$ and $\mathcal{Y}$ be two topological spaces, and let $U \subset \mathcal{X}$ and $V \subset \mathcal{Y}$ be open subsets. Consider the family of all cartesian products of the form $U \times V$. The topology formed from these products of open subsets is called the product topology for $\mathcal{X} \times \mathcal{Y}$. If $W \subset \mathcal{X} \times \mathcal{Y}$, then $W$ is open relative to the product topology iff for each point $(x, y) \in \mathcal{X} \times \mathcal{Y}$ there are open neighborhoods, $U$ of $x$ and $V$ of $y$, such that $U \times V \subset W$. For example, the usual topology for $d$-dimensional Euclidean space $\Re^{d}$ consists of all open sets of points in $\Re^{d}$, and this topology is equivalent to the product topology for the product of $d$ copies of $\Re$.

One of the core elements of manifold learning involves the idea of “embedding” one topological space inside another. Loosely speaking, the space $\mathcal{X}$ is said to be embedded in the space $\mathcal{Y}$ if the topological properties of $\mathcal{Y}$ when restricted to $\mathcal{X}$ are identical to the topological properties of $\mathcal{X}$. To be more specific, we state the following definitions. A function $g: \mathcal{X} \rightarrow \mathcal{Y}$ is said to be continuous if the inverse image of an open set in $\mathcal{Y}$ is an open set in $\mathcal{X}$. If $g$ is a bijective (i.e., one-to-one and onto) function such that $g$ and its inverse $g^{-1}$ are continuous, then $g$ is said to be a homeomorphism. Two topological spaces $\mathcal{X}$ and $\mathcal{Y}$ are said to be homeomorphic (or topologically equivalent) if there exists a homeomorphism from one space onto the other. A topological space $\mathcal{X}$ is said to be embedded in a topological space $\mathcal{Y}$ if $\mathcal{X}$ is homeomorphic to a subspace of $\mathcal{Y}$.

If $A \subset \mathcal{X}$, then $A$ is said to be compact if every class of open sets whose union contains $A$ has a finite subclass whose union also contains $A$ (i.e., if every open cover of $A$ contains a finite subcover). This definition of compactness extends naturally to the topological space $\mathcal{X}$, and is itself a generalization of the celebrated Heine-Borel theorem that says that closed and bounded subsets of $\Re$ are compact. We note that subsets of a compact space need not be compact; however, closed subsets will be compact. Tychonoff’s theorem that the product of compact spaces is compact is said to be “probably the most important single theorem of general topology” (Kelley, 1955, p. 143). One of the properties of compact spaces is that if $g: \mathcal{X} \rightarrow \mathcal{Y}$ is continuous and $\mathcal{X}$ is compact, then $g(\mathcal{X})$ is a compact subspace of $\mathcal{Y}$.

Another important idea in topology is that of a connected space. A topological space $\mathcal{X}$ is said to be connected if it cannot be represented as the union of two disjoint, nonempty, open sets. For example, $\Re$ itself with the usual topology is a connected space, and an interval in $\Re$ containing at least two points is connected. Furthermore, if $g: \mathcal{X} \rightarrow \mathcal{Y}$ is continuous and $\mathcal{X}$ is connected, then its image, $g(\mathcal{X})$, is connected as a subspace of $\mathcal{Y}$. Also, the product of any number of nonempty connected spaces, such as $\Re^{d}$ for any $d \geq 1$, is connected. The space $\mathcal{X}$ is disconnected if it is not connected.

A topological space $\mathcal{X}$ is said to be locally Euclidean if there exists an integer $d \geq 0$ such that around every point in $\mathcal{X}$, there is a local neighborhood which is homeomorphic to an open subset in Euclidean space $\Re^{d}$. A topological space $\mathcal{X}$ is a Hausdorff space if every pair of distinct points has a corresponding pair of disjoint neighborhoods. Almost all spaces are Hausdorff, including the real line $\Re$ with the standard metric topology. Also, subspaces and products of Hausdorf spaces are Hausdorff. $\mathcal{X}$ is second-countable if its topology has a countable basis of open sets. Most reasonable topological spaces are second countable, including the real line $\Re$, where the usual topology of open intervals has rational numbers as interval endpoints; a finite product of $\Re$ with itself is second countable if its topology is the product topology where open intervals have rational endpoints. Subspaces of second-countable spaces are again second countable.

机器学习代写|流形学习代写manifold data learning代考|Spectral Embedding Methods for Manifold Learning


机器学习代写|流形学习代写manifold data learning代考|Alan Julian Izenman

流形学习涵盖了几何、计算和统计学的大部分学科,已成为数据挖掘和统计学习的重要研究课题。流形学习最简单的描述是它是一类用于恢复嵌入在高维环境空间中的低维流形的算法。在恢复高维数据的低维非线性嵌入的方法上的重大突破(Tenenbaum、de Silva 和 Langford,2000;Roweis 和 Saul,2000)导致构建了许多其他用于执行非线性流形学习的算法及其关闭相对的,非线性的降维。所有嵌入算法的主要工具是与适当随机矩阵的顶部几个或底部几个特征值相关联的特征向量集。我们将这些算法称为谱嵌入方法。谱嵌入方法旨在恢复线性或非线性流形,通常在高维空间中。

长期以来,线性方法一直被认为是统计学家工具箱的重要组成部分,包括主成分分析 (PCA) 和多维缩放 (MDS)。PCA 已成功用于许多不同的学科和应用。例如,在计算机视觉中,PCA 用于研究形状、外观和运动的抽象概念,以帮助解决面部和物体识别、监视、人员跟踪、安全和图像压缩中的高维数据问题(Turk 和彭特兰,1991 年;德拉托雷和布莱克,2001 年)。在天文学中,超大型数字巡天已成为常态,PCA 已被用于分析和分类恒星光谱,对星系和类星体进行形态和光谱分类,以及分析超新星遗迹的图像(Steiner、Menezes、Ricci 和奥利维拉,2009)。在生物信息学中,PCA 已被用于研究由对各种组织来源的全基因组基因表达实验产生的高维数据,其中此类研究中主要主要成分的散点图通常显示特定类别的基因,这些基因由不同的具有独特生物学特征的集群(Yeung 和 Ruzzo,2001;ZhengBradley、Rung、Parkinson 和 Brazma,2010)。PCA 还被用于选择单核苷酸多态性 (SNP) 的最佳子集 (Lin and Altman, 2004)。PCA 也是 其中,此类研究中主要主成分的散点图通常显示特定类别的基因,这些基因由不同的独特生物学特征簇表达(Yeung 和 Ruzzo,2001;ZhengBradley、Rung、Parkinson 和 Brazma,2010)。PCA 还被用于选择单核苷酸多态性 (SNP) 的最佳子集 (Lin and Altman, 2004)。PCA 也是 其中,此类研究中主要主成分的散点图通常显示特定类别的基因,这些基因由不同的独特生物学特征簇表达(Yeung 和 Ruzzo,2001;ZhengBradley、Rung、Parkinson 和 Brazma,2010)。PCA 还被用于选择单核苷酸多态性 (SNP) 的最佳子集 (Lin and Altman, 2004)。PCA 也是


MDS 起源于心理学,最近被发现在生物信息学中最有用,它被称为“距离几何”。例如,MDS 已被用于显示蛋白质结构宇宙的全局表示(即地图)(Holm 和 Sander,1996;Hou、Sims、Zhang 和 Kim,2003;Hou、Jun、Zhang 和Kim,2005;Lu、Keles、Wright 和 Wahba,2005;Kim、Ahn、Lee、Park 和 Kim,2010)。这个想法是,与其他点紧密定位的点提供了有关同一家族中蛋白质形状和功能的重要信息,因此可用于预测和分类目的。有关 MDS 中许多不同应用领域和研究主题的列表,请参见 Izenman (2008, Table 13.1)。

机器学习代写|流形学习代写manifold data learning代考|Spaces and Manifolds

流形学习涉及来自一般拓扑和微分几何的概念。对拓扑空间的良好介绍包括 Kelley (1955)、Willard (1970)、Bourbaki (1989)、Mendelson (1990)、Steen (1995)、James (1999),其中一些已被重印。有关微分几何的书籍包括 Spivak (1965)、Kreyszig (1991)、Kühnel (2000)、Lee (2002) 和 Pressley (2010)。


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

Maurice Fréchet (1906) 引入了拓扑空间(以度量空间的形式),这个想法在接下来的几十年中得到发展和扩展。对这个主题做出重大贡献的人中有 Felix Hausdorff,他在 1914 年使用 Johann Benedict Listing 的德语单词 Topologie 创造了“拓扑空间”一词。1847.

拓扑空间X是子集的非空集合X它包含空集、空间本身以及这些集合的任意并集和有限交集。拓扑空间通常表示为(X,吨), 在哪里吨表示与相关的拓扑X. 的元素吨被称为开集X, 如果它的补集是开集,则它是闭集。拓扑空间也可以通过邻域的概念来表征。如果X是拓扑空间中的一个点X, 它的邻域是一个包含一个开集的集合

让X和是是两个拓扑空间,令在⊂X和在⊂是是开放子集。考虑以下形式的所有笛卡尔积的族在×在. 由这些开放子集的乘积形成的拓扑称为乘积拓扑X×是. 如果在⊂X×是, 然后在对于每个点相对于产品拓扑是开放的(X,是)∈X×是有开放的社区,在的X和在的是, 这样在×在⊂在. 例如,通常的拓扑d维欧几里得空间ℜd由所有开集的点组成ℜd, 这个拓扑等价于 的乘积的乘积拓扑d的副本ℜ.

流形学习的核心要素之一涉及将一个拓扑空间“嵌入”另一个拓扑空间的想法。说白了就是空间X据说嵌入空间是如果拓扑性质是当限制在X与拓扑性质相同X. 更具体地说,我们陈述以下定义。一个函数G:X→是如果一个开集的逆像在是是一个开集X. 如果G是一个双射(即,一对一和上)函数,使得G和它的逆G−1是连续的,那么G据说是同胚。两个拓扑空间X和是如果存在从一个空间到另一个空间的同胚,则称其是同胚的(或拓扑等价的)。拓扑空间X据说嵌入在拓扑空间中是如果X同胚于一个子空间是.

如果一种⊂X, 然后一种如果每类开集的并集包含一种有一个有限子类,其并集也包含一种(即,如果每个打开的盖子一种包含一个有限的子覆盖)。这种紧致性的定义自然地延伸到拓扑空间X,并且它本身就是著名的 Heine-Borel 定理的推广,该定理说ℜ紧凑。我们注意到紧致空间的子集不一定是紧致的;但是,封闭子集将是紧凑的。Tychonoff 的紧致空间的乘积是紧致的定理被称为“可能是一般拓扑中最重要的单一定理”(Kelley,1955,第 143 页)。紧致空间的性质之一是,如果G:X→是是连续的并且X是紧致的,那么G(X)是一个紧致子空间是.

拓扑学中的另一个重要思想是连通空间。拓扑空间X如果它不能表示为两个不相交的非空开集的并集,则称它是连通的。例如,ℜ具有通常拓扑的自身是一个连通空间,而在ℜ至少包含两个点是相连的。此外,如果G:X→是是连续的并且X是连接的,然后是它的图像,G(X), 连接为是. 此外,任意数量的非空连通空间的乘积,例如ℜd对于任何d≥1, 已连接。空间X如果未连接,则断开连接。

拓扑空间X如果存在整数,则称其为局部欧几里得d≥0这样在每个点周围X,有一个局部邻域同胚于欧几里得空间中的一个开子集ℜd. 拓扑空间X是 Hausdorff 空间,如果每对不同的点都有对应的一对不相交的邻域。几乎所有空间都是豪斯多夫,包括实线ℜ使用标准度量拓扑。同样,豪斯多夫空间的子空间和乘积是豪斯多夫。X如果它的拓扑具有开集的可数基,则它是第二可数的。最合理的拓扑空间是秒可数的,包括实线ℜ,其中通常的开区间拓扑以有理数作为区间端点;的有限乘积ℜ如果它的拓扑是开区间有理性端点的乘积拓扑,with 本身是第二可数的。秒可数空间的子空间又是秒可数的。

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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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