### 机器学习代写|流形学习代写manifold data learning代考|Nonlinear 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|Nonlinear Manifold Learning

We next discuss some algorithmic techniques that proved to be innovative in the study of nonlinear manifold learning: ISOMAP, LOCAL LINEAR EMBEDDING, LAPLACIAN EIGENMAPS, DIFFUSION MAPS, HESSIAN EIGENMAPS, and the many different versions of NONLINEAR PCA. The goal of each of these algorithms is to recover the full low-dimensional

representation of an unknown nonlinear manifold, $\mathcal{M}$, embedded in some high-dimensional space, where it is important to retain the neighborhood structure of $\mathcal{M}$. When $\mathcal{M}$ is highly nonlinear, such as the S-shaped manifold in the left panel of Figure $1.1$, these algorithms outperform the usual linear techniques. The nonlinear manifold-learning methods emphasize simplicity and avoid optimization problems that could produce local minima.

Assume that we have a finite random sample of data points, $\left{\mathbf{y}{i}\right}$, from a smooth $t$ dimensional manifold $\mathcal{M}$ with metric given by the geodesic distance $d^{\mathcal{M}} ;$ see Section $1.2 .4$. These points are then nonlinearly embedded by a smooth map $\psi$ into high-dimensional input space $\mathcal{X}=\Re^{r}(t \ll r)$ with Euclidean metric $|$. $| \mathcal{X}$. This embedding provides us with the input data $\left{\mathbf{x}{i}\right}$. For example, in the right panel of Figure $1.1$, we randomly generated 20,000 three-dimensional points to lie uniformly on the surface of the two-dimensional Sshaped curve displayed in the left panel. Thus, $\psi: \mathcal{M} \rightarrow \mathcal{X}$ is the embedding map, and a point on the manifold, $\mathbf{y} \in \mathcal{M}$, can be expressed as $\mathbf{y}=\phi(\mathbf{x}), \mathbf{x} \in \mathcal{X}$, where $\phi=\psi^{-1}$. The goal is to recover $\mathcal{M}$ and find an implicit representation of the map $\psi$ (and, hence, recover the $\left.\left{\mathbf{y}{i}\right}\right)$, given only the input data points $\left{\mathbf{x}{i}\right}$ in $\mathcal{X}$.

Each algorithm computes $t^{\prime}$-dimensional estimates, $\left{\widehat{\mathbf{y}}{i}\right}$, of the $t$-dimensional manifold data, $\left{\mathbf{y}{i}\right}$, for some $t^{\prime}$. Such a reconstruction is deemed to be successful if $t^{\prime}=t$, the true (unknown) dimensionality of $\mathcal{M}$. In practice, $t^{\prime}$ will most likely be too large. Because we require a low-dimensional solution, we retain only the first two or three of the coordinate vectors and plot the corresponding elements of those vectors against each other to yield $n$ points in two- or three-dimensional space. For all practical purposes, such a display is usually sufficient to identify the underlying manifold.

Most of the nonlinear manifold-learning algorithms that we discuss here are based upon different philosophies regarding how one should recover unknown nonlinear manifolds. However, they each consist of a three-step approach (except NONLINEAR PCA). The first and third steps are common to all algorithms: the first step incorporates neighborhood information at each data point to construct a weighted graph having the data points as vertices, and the third step is a spectral embedding step that involves an $(n \times n)$-eigenequation computation. The second step is specific to the algorithm, taking the weighted neighborhood graph and transforming it into suitable input for the spectral embedding step.

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

The isometric feature mapping (or IsOMAP) algorithm (Tenenbaum, de Silva, and Langford, 2000 ) assumes that the smooth manifold $\mathcal{M}$ is a convex region of $\Re^{t}(t \ll r)$ and that the embedding $\psi: \mathcal{M} \rightarrow \mathcal{X}$ is an isometry. This assumption has two key ingredients:

• Isometry: The geodesic distance is invariant under the map $\psi$. For any pair of points on the manifold, $\mathbf{y}, \mathbf{y}^{\prime} \in \mathcal{M}$, the geodesic distance between those points equals the Euclidean distance between their corresponding coordinates, $\mathbf{x}, \mathbf{x}^{\prime} \in \mathcal{X}$; i.e.,
$$d^{\mathcal{M}}\left(\mathbf{y}, \mathbf{y}^{\prime}\right)=\left|\mathbf{x}-\mathbf{x}^{\prime}\right|_{\mathcal{X}}$$
where $\mathbf{y}=\phi(\mathbf{x})$ and $\mathbf{y}^{\prime}=\phi\left(\mathbf{x}^{\prime}\right)$.
• Convexity: The manifold $\mathcal{M}$ is a convex subset of $\Re^{t}$.
IsomaP considers $\mathcal{M}$ to be a convex region possibly distorted in any of a number of ways (e.g., by folding or twisting). The so-called Swiss roll, ${ }^{2}$ which is a flat two-dimensional rectangular submanifold of $\Re^{3}$, is one such example; see Figure $1.2$. Empirical studies show that IsOMAP works well for intrinsically flat submanifolds of $\mathcal{X}=\Re^{r}$ that look like rolledup sheets of paper or “open” manifolds such as an open box or open cylinder. However, IsOMAP does not perform well if there are any holes in the roll, because this would violate the convexity assumption. The isometry assumption appears to be reasonable for certain types of situations, but, in many other instances, the convexity assumption may be too restrictive (Donoho and Grimes, 2003b).

IsomAP uses the isometry and convexity assumptions to form a nonlinear generalization of multidimensional scaling (MDS). Recall that MDS looks for a low-dimensional subspace in which to embed input data while preserving the Euclidean interpoint distances (see Section 1.3.2). Unfortunately, working with Euclidean distances in MDS when dealing with curved regions tends to give poor results. IsOMAP follows the general MDS philosophy by attempting to preserve the global geometric properties of the underlying nonlinear manifold, and it does this by approximating all pairwise geodesic distances (i.e., lengths of the shortest paths between two points) on the manifold. In this sense, IsOMAP provides a global approach to manifold learning.

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

The Laplacian eigenmap algorithm (Belkin and Niyogi, 2002) also consists of three steps. The first and third steps of the Laplacian eigenmap algorithm are very similar to the first and third steps, respectively, of the LLE algorithm.

1. Nearest-neighbor search. Fix an integer $K$ or an $\epsilon>0$. The neighborhoods of each data point are symmetrically defined: for a $K$-neighborhood $N_{i}^{K}$ of the point $\mathbf{x}{i}$, let $\mathbf{x}{j} \in N_{i}^{K}$ iff $\mathbf{x}{i} \in N{j}^{K}$; similarly, for an $\epsilon$-neighborhood $N_{i}^{\epsilon}$, let $\mathbf{x}{j} \in N{i}^{e}$ iff $\left|\mathbf{x}{i}-\mathbf{x}{j}\right|<\epsilon$, where the norm is Euclidean norm. In general, let $N_{i}$ denote the neighborhood of $\mathbf{x}_{i}$.
2. Weighted adjacency matrix. Let $\mathbf{W}=\left(w_{i j}\right)$ be a symmetric $(n \times n)$ weighted adjacency matrix defined as follows:
$$w_{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{j}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { otherwise }\end{cases}$$

These weights are determined by the isotropic Gaussian kernel (also known as the heat kernel), with scale parameter $\sigma$. Denote the resulting weighted graph by $\mathcal{G}$. If $\mathcal{G}$ is not connected, apply step 3 to each connected subgraph.

1. Spectral embedding. Let $\mathbf{D}=\left(d_{i j}\right)$ be an $(n \times n)$ diagonal matrix with diagonal elements $d_{i i}=\sum_{j \in N_{i}} w_{i j}=\left(\mathbf{W} 1_{n}\right){i}, i=1,2, \ldots, n$. The $(n \times n)$ symmetric matrix $\mathbf{L}=\mathbf{D}-\mathbf{W}$ is known as the graph Laplacian for the graph $\mathcal{G}$. Let $\mathbf{y}=\left(y{i}\right)$ be an $n$-vector. Then, $\mathbf{y}^{\tau} \mathbf{L} \mathbf{y}=\frac{1}{2} \sum_{i=1}^{n} \sum_{j=1}^{n} w_{i j}\left(y_{i}-y_{j}\right)^{2}$, so that $\mathbf{L}$ is nonnegative definite.

When data are uniformly sampled from a low-dimensional manifold $\mathcal{M}$ of $\Re^{r}$, the graph Laplacian $\mathbf{L}=\mathbf{L}{n, \sigma}$ (considered as a function of $n$ and $\sigma$ ) can be regarded as a discrete approximation to the continuous Laplace-Beltrami operator $\Delta{\mathcal{M}}$ defined on the manifold $\mathcal{M}$, and converges to $\Delta_{\mathcal{M}}$ as $\sigma \rightarrow 0$ and $n \rightarrow \infty$. Furthermore, when the data are sampled from an arbitrary probability distribution $P$ on the manifold $\mathcal{M}$, then, under certain conditions on $\mathcal{M}$ and $P$, the graph Laplacian converges to a weighted version of $\Delta_{\mathcal{M}}$ (Belkin and Niyogi, 2008).

The $(t \times n)$-matrix $\mathbf{Y}=\left(\mathbf{y}{1}, \cdots, \mathbf{y}{n}\right)$, which is used to embed the graph $\mathcal{G}$ into the low-dimensional space $\Re^{t}$, where $\mathbf{y}{i}$ yields the embedding coordinates of the $i$ th point, is determined by minimizing the objective function, $$\sum{i} \sum_{j} w_{i j}\left|\mathbf{y}{i}-\mathbf{y}{j}\right|^{2}=\operatorname{tr}\left{\mathbf{Y L Y} \mathbf{Y}^{\tau}\right}$$
In other words, we seek the solution,
$$\widehat{\mathbf{Y}}=\arg \min {\mathbf{Y}: \mathbf{Y D \mathbf { Y } ^ { \top } = \mathbf { I } { t }}} \operatorname{tr}\left{\mathbf{Y L Y ^ { \top } }},\right.$$
where we restrict $\mathbf{Y}$ such that $\mathbf{Y D Y} \mathbf{Y}^{\tau}=\mathbf{I}{t}$ to prevent a collapse onto a subspace of fewer than $t-1$ dimensions. The solution is given by the generalized eigenequation, $\mathbf{L v}=\lambda \mathbf{D v}$, or, equivalently, by finding the eigenvalues and eigenvectors of the matrix $\widehat{\mathbf{W}}=\mathbf{D}^{-1 / 2} \mathbf{W D} \mathbf{D}^{-1 / 2}$. The smallest eigenvalue, $\lambda{n}$, of $\widehat{\mathbf{W}}$ is zero. If we ignore the smallest eigenvalue (and its corresponding constant eigenvector $\mathbf{v}{n}=1{n}$ ), then the best embedding solution in $\Re^{t}$ is similar to that given by LLE; that is, the rows of $\widehat{\mathbf{Y}}$ are the eigenvectors,
$$\widehat{\mathbf{Y}}=\left(\widehat{\mathbf{y}}{1}, \cdots, \widehat{\mathbf{y}}{n}\right)=\left(\mathbf{v}{n-1}, \cdots, \mathbf{v}{n-t}\right)^{\tau},$$
corresponding to the next $t$ smallest eigenvalues, $\lambda_{n-1} \leq \cdots \leq \lambda_{n-t}$, of $\widehat{\mathbf{W}}$.

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

• 等距：测地线距离在地图下是不变的ψ. 对于流形上的任意一对点，是,是′∈米，这些点之间的测地线距离等于它们对应坐标之间的欧几里得距离，X,X′∈X; IE，
d米(是,是′)=|X−X′|X
在哪里是=φ(X)和是′=φ(X′).
• 凸性：流形米是一个凸子集ℜ吨.
IsomaP 认为米成为可能以多种方式（例如，通过折叠或扭曲）中的任何一种方式扭曲的凸面区域。所谓的瑞士卷，2它是一个平面二维矩形子流形ℜ3, 就是这样一个例子；见图1.2. 实证研究表明，IsOMAP 适用于本质平坦的子流形X=ℜr看起来像卷起的纸或“打开”的歧管，例如打开的盒子或打开的圆柱体。但是，如果卷中有任何孔洞，IsOMAP 的性能就不会很好，因为这会违反凸性假设。对于某些类型的情况，等距假设似乎是合理的，但在许多其他情况下，凸性假设可能过于严格（Donoho 和 Grimes，2003b）。

IsomAP 使用等距和凸性假设来形成多维尺度 (MDS) 的非线性泛化。回想一下，MDS 寻找一个低维子空间，在其中嵌入输入数据，同时保留欧几里德点间距离（参见第 1.3.2 节）。不幸的是，在处理弯曲区域时，在 MDS 中使用欧几里德距离往往会产生较差的结果。IsOMAP 遵循一般 MDS 哲学，试图保留底层非线性流形的全局几何特性，它通过近似流形上的所有成对测地线距离（即两点之间的最短路径的长度）来做到这一点。从这个意义上说，IsOMAP 为流形学习提供了一种全局方法。

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

1. 最近邻搜索。修复一个整数ķ或一个ε>0. 每个数据点的邻域是对称定义的：对于ķ-邻里ñ一世ķ点的X一世， 让Xj∈ñ一世ķ当且当X一世∈ñjķ; 同样，对于一个ε-邻里ñ一世ε， 让Xj∈ñ一世和当且当|X一世−Xj|<ε，其中范数是欧几里得范数。一般来说，让ñ一世表示邻域X一世.
2. 加权邻接矩阵。让在=(在一世j)是对称的(n×n)加权邻接矩阵定义如下：
w_{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{ x}{i}-\mathbf{x}{j}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \在 N{i} \ 0, & \text { 否则 }\end{cases}w_{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{ x}{i}-\mathbf{x}{j}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \在 N{i} \ 0, & \text { 否则 }\end{cases}

1. 光谱嵌入。让D=(d一世j)豆(n×n)具有对角元素的对角矩阵d一世一世=∑j∈ñ一世在一世j=(在1n)一世,一世=1,2,…,n. 这(n×n)对称矩阵大号=D−在被称为图 Laplacian for the graphG. 让是=(是一世)豆n-向量。然后，是τ大号是=12∑一世=1n∑j=1n在一世j(是一世−是j)2， 以便大号是非负定的。

\widehat{\mathbf{Y}}=\arg \min {\mathbf{Y}: \mathbf{Y D \mathbf { Y } ^ { \top } = \mathbf { I } { t }}} \operatorname{tr }\left{\mathbf{Y L Y ^ { \top } }},\right.\widehat{\mathbf{Y}}=\arg \min {\mathbf{Y}: \mathbf{Y D \mathbf { Y } ^ { \top } = \mathbf { I } { t }}} \operatorname{tr }\left{\mathbf{Y L Y ^ { \top } }},\right.

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