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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• 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|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|Alan Julian Izenman

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|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.

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