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

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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|Diffusion Maps

The basic idea of Diffusion MaPs (Nadler, Lafon, Coifman, and Kevrekidis, 2005; Coifman and Lafon, 2006) uses a Markov chain constructed over a graph of the data points, followed by an eigenanalysis of the probability transition matrix of the Markov chain. As with the other algorithms in this Section, there are three steps in this algorithm, with the first and second steps the same as for Laplacian eigenmaps. Although a nearest-neighbor search (Step 1) was not explicitly considered in the above papers on diffusion maps as a means of constructing the graph (Step 2), a nearest-neighbor search is included in software packages for computing diffusion maps. For an example in astronomy of a diffusion map incorporating a nearest-neighbor search, see Freeman, Newman, Lee, Richards, and Schafer (2009).

1. Nearest-Neighbor Search. Fix an integer $K$ or an $\epsilon>0$. Define a $K$-neighborhood $N_{i}^{K}$ or an $\epsilon$-neighborhood $N_{i}^{e}$ of the point $\mathbf{x}{i}$ as in Step 1 of Laplacian eigenmaps. In general, let $N{i}$ denote the neighborhood of $\mathbf{x}_{i}$.Pairwise Adjacency Matrix. The $n$ data points $\left{\mathbf{x}{i}\right}$ in $\Re^{r}$ can be regarded as a graph $\mathcal{G}=\mathcal{G}(\mathcal{V}, \mathcal{E})$ with the data points playing the role of vertices $\mathcal{V}=\left{\mathbf{x}{1}, \ldots, \mathbf{x}{n}\right}$, and the set of edges $\mathcal{E}$ are the connection strengths (or weights), $w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)$, between pairs of adjacent vertices, $$w{i j}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)= \begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{i}\right|^{2}}{2 \sigma^{2}}\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { otherwise. }\end{cases} 2.$$
3. This is a Gaussian kernel with width $\sigma$; however, other kernels may be used. Kernels such as (1.52) ensure that the closer two points are to each other, the larger the value of $w$. For convenience in exposition, we will suppress the fact that the elements of most of the matrices depend upon the value of $\sigma$. Then, $\mathbf{W}=\left(w_{i j}\right)$ is a pairwise adjacency matrix between the $n$ points. To make the matrix $\mathbf{W}$ even more sparse, values of its entries that are smaller than some given threshold (i.e., the points in question are far apart from each other) can be set to zero. The graph $\mathcal{G}$ with weight matrix W gives information on the local geometry of the data.
4. Spectral embedding. Define $\mathbf{D}=\left(d_{i j}\right)$ to be a diagonal matrix formed from the matrix W by setting the diagonal elements, $d_{i i}=\sum_{j} w_{i j}$, to be the column sums of $\mathbf{W}$ and the off-diagonal elements to be zero. The $(n \times n)$ symmetric matrix $\mathbf{L}=\mathbf{D}-\mathbf{W}$ is the graph Laplacian for the graph $\mathcal{G}$. We are interested in the solutions of the generalized eigenequation, $\mathbf{L v}=\lambda \mathbf{D v}$, or, equivalently, of the matrix
5. $$6. \mathbf{P}=\mathbf{D}^{-1 / 2} \mathbf{L} \mathbf{D}^{-1 / 2}=\mathbf{I}_{n}-\mathbf{D}^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2}, 7.$$
8. which is the normalized graph Laplacian. The matrix $\mathbf{H}=e^{t \mathbf{P}}, t \geq 0$, is usually referred to as the heat kernel. By construction, $\mathbf{P}$ is a stochastic matrix with all row sums equal to one, and, thus, can be interpreted as defining a random walk on the graph $\mathcal{G}$.

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

Recall that, in certain situations, the convexity assumption for IsOMAP may be too restrictive. Instead, we may require that the manifold $\mathcal{M}$ be locally isometric to an open, connected subset of $\Re^{t}$. Popular examples include families of “articulated” images (i.e., translated or rotated images of the same object, possibly through time) that are found in a high-dimensional, digitized-image library (e.g., faces, pictures, handwritten numbers or letters). However, if the pixel elements of each 64 -pixel-by-64-pixel digitized image are represented as a 4,096 -dimensional vector in “pixel space,” it would be very difficult to show that the images really live on a low-dimensional manifold, especially if that image manifold is unknown.

We can model such images using a vector of smoothly varying articulation parameters $\boldsymbol{\theta} \in \boldsymbol{\Theta}$. For example, digitized images of a person’s face that are varied by pose and illumination can be parameterized by two pose parameters (expression [happy, sad, sleepy, surprised, wink] and glasses-no glasses) and a lighting direction (centerlight, leftlight, rightlight, normal); similarly, handwritten ” 2 “s appear to be parameterized essentially by two features, bottom loop and top arch (Tenenbaum, de Silva, and Langford, 2000; Roweis and Saul, 2000). To some extent, learning about an underlying image manifold depends upon whether the images are sufficiently scattered around the manifold and how good is the quality of digitization of each image?

HESSIAN EIGENMAPS (Donoho and Grimes, 2003b) were proposed for recovering manifolds of high-dimensional libraries of articulated images where the convexity assumption is often violated. Let $\Theta \subset \Re^{t}$ be the parameter space and suppose that $\phi: \Theta \rightarrow R^{r}$, where $t<r$. Assume $\mathcal{M}=\phi(\Theta)$ is a smooth manifold of articulated images. The isometry and convexity requirements of IsoMAP are replaced by the following weaker requirements:

• Local Isometry: $\phi$ is a locally isometric embedding of $\Theta$ into $\Re^{r}$. For any point $\mathbf{x}^{\prime}$ in a sufficiently small neighborhood around each point $x$ on the manifold $\mathcal{M}$, the geodesic distance equals the Euclidean distance between their corresponding parameter points $\boldsymbol{\theta}, \boldsymbol{\theta}^{\prime} \in \Theta ;$ that is,
$$d^{M}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\left|\theta-\theta^{\prime}\right|_{\Theta+}$$
where $\mathbf{x}=\phi(\boldsymbol{\theta})$ and $\mathbf{x}^{\prime}=\phi\left(\boldsymbol{\theta}^{\prime}\right)$
• Connectedness: The parameter space $\theta$ is an open, connected subset of $\Omega^{t}$.
The goal is to recover the parameter vector $\boldsymbol{\theta}$ (up to a rigid motion).

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

Another way of dealing with nonlinear manifold learning is to construct nonlinear versions of linear manifold learning techniques. We have already seen how Isomap provides a nonlinear generalization of MDS. How can we generalize PCA to the nonlinear case? In this Section,

we briefly describe the basic ideas behind POLYNOMIAL PCA, PRINCIPAL CuRVES AND SURFACES, MULTILAYER AUTOASSOCIATIVE NEURAL NETWORKS, and KerneL PCA.
Polynomial PCA
There have been several different attempts to generalize PCA to data living on or near nonlinear manifolds of a lower-dimensional space than input space. The first such idea was to add to the set of $r$ input variables quadratic, cubic, or higher-degree polynomial transformations of those input variables, and then apply linear PCA. The result is POLYNOMIAL PCA (Gnanadesikan and Wilk, 1969), whose embedding coordinates are the eigenvectors corresponding to the smallest few eigenvalues of the expanded covariance matrix.

In the original study of polynomial PCA, the method was illustrated with a quadratic transformation of bivariate input variables. In this scenario, $\left(X_{1}, X_{2}\right)$ expands to become $\left(X_{1}, X_{2}, X_{1}^{2}, X_{2}^{2}, X_{1} X_{2}\right)$. This formulation is feasible, but for larger problems, the possibilities become more complicated. First, the variables in the expanded set will not be scaled in a uniform manner, so that standardization will be necessary, and second, the number of variables in the expanded set will increase rapidly with large $r$, which will lead to bigger computational problems. Gnanadesikan and Wilk’s article, however, gave rise to a variety of attempts to define a more general nonlinear version of PCA.
Principal Curves and Surfaces
The next attempt at creating a nonlinear PCA was PRINCIPAL CURVES AND SURFACES (Hastie, 1984; Hastie and Stuetzle, 1989). A principal curve is a smooth one-dimensional curve that passes through the “middle” of the data, and a principal surface (or principal manifold) is a generalization of a principal curve to a smooth two- or higher-dimensional manifold. So, we can visualize principal curves and surfaces as defining a nonlinear manifold in higher-dimensional input space.

Let $\mathbf{x} \in \Re^{r}$ be a data point and let $\mathbf{f}(\lambda)$ be a curve, $\lambda \in \Lambda$; see Section $1.2 .4$ for definitions. Project $\mathbf{x}$ to a point on $\mathbf{f}(\lambda)$ that is closest in Euclidean distance to $\mathbf{x}$. Define the projection index
$$\lambda_{\mathbf{f}}(\mathbf{x})=\sup {\lambda}\left{\lambda:|\mathbf{x}-\mathbf{f}(\lambda)|=\inf {\mu}|\mathbf{x}-\mathbf{f}(\mu)|\right}$$

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

Diffusion Maps 的基本思想（Nadler、Lafon、Coifman 和 Kevrekidis，2005 年；Coifman 和 Lafon，2006 年）使用在数据点图上构建的马尔可夫链，然后对马尔可夫链的概率转移矩阵进行特征分析. 与本节中的其他算法一样，该算法有三个步骤，第一步和第二步与拉普拉斯特征图相同。尽管在上述关于扩散图的论文中没有明确考虑最近邻搜索（步骤 1）作为构建图的一种手段（步骤 2），但最近邻搜索包含在用于计算扩散图的软件包中。有关包含最近邻搜索的扩散图的天文学示例，请参见 Freeman、Newman、Lee、Richards 和 Schafer (2009)。

1. 最近邻搜索。修复一个整数ķ或一个ε>0. 定义一个ķ-邻里ñ一世ķ或一个ε-邻里ñ一世和点的X一世如拉普拉斯特征图的第 1 步。一般来说，让ñ一世表示邻域X一世.成对邻接矩阵。这n数据点\left{\mathbf{x}{i}\right}\left{\mathbf{x}{i}\right}在ℜr可以看成图G=G(在,和)数据点扮演顶点的角色\mathcal{V}=\left{\mathbf{x}{1}, \ldots, \mathbf{x}{n}\right}\mathcal{V}=\left{\mathbf{x}{1}, \ldots, \mathbf{x}{n}\right}, 和边的集合和是连接强度（或权重），在(X一世,Xj)，在相邻顶点对之间，$$w{ij}=w\left(\mathbf{x}{i}, \mathbf{x}{j}\right)=\begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{i}\right|^{2}}{2 \sigma^{2} }\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { 否则。}\结束{案例}\begin{cases}\exp \left{-\frac{\left|\mathbf{x}{i}-\mathbf{x}{i}\right|^{2}}{2 \sigma^{2} }\right}, & \text { if } \mathbf{x}{j} \in N{i} \ 0, & \text { 否则。}\结束{案例} 2.$$
3. 这是一个具有宽度的高斯核σ; 但是，可以使用其他内核。（1.52）等内核保证两点距离越近，值越大在. 为方便说明，我们将隐藏大多数矩阵的元素取决于σ. 然后，在=(在一世j)是之间的成对邻接矩阵n点。制作矩阵在甚至更稀疏，其条目的值小于某个给定阈值（即，所讨论的点彼此相距很远）可以设置为零。图表G权重矩阵 W 给出了数据的局部几何信息。
4. 光谱嵌入。定义D=(d一世j)是通过设置对角元素由矩阵 W 形成的对角矩阵，d一世一世=∑j在一世j, 为的列总和在和非对角线元素为零。这(n×n)对称矩阵大号=D−在是图的拉普拉斯算子G. 我们对广义特征方程的解感兴趣，大号在=λD在, 或者, 等价的, 矩阵
5. $$6. \mathbf{P}=\mathbf{D}^{-1 / 2} \mathbf{L} \mathbf{D}^{-1 / 2}=\mathbf{I}_{n}-\mathbf{D }^{-1 / 2} \mathbf{W} \mathbf{D}^{-1 / 2}, 7.$$
8. 这是归一化图拉普拉斯算子。矩阵H=和吨磷,吨≥0, 通常称为热核。通过施工，磷是一个随机矩阵，所有行和都等于 1，因此可以解释为在图上定义随机游走G.

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

HESSIAN EIGENMAPS (Donoho and Grimes, 2003b) 被提出用于恢复经常违反凸性假设的铰接图像的高维库的流形。让θ⊂ℜ吨是参数空间并假设φ:θ→Rr， 在哪里吨<r. 认为米=φ(θ)是铰接图像的平滑流形。IsoMAP 的等距和凸度要求被以下较弱的要求取代：

• 局部等距：φ是一个局部等距嵌入θ进入ℜr. 对于任何一点X′在每个点周围足够小的邻域中X在歧管上米，测地线距离等于它们对应的参数点之间的欧几里得距离θ,θ′∈θ;那是，
d米(X,X′)=|θ−θ′|θ+
在哪里X=φ(θ)和X′=φ(θ′)
• 连通性：参数空间θ是一个开放的、连通的子集Ω吨.
目标是恢复参数向量θ（直到刚性运动）。

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

\lambda_{\mathbf{f}}(\mathbf{x})=\sup {\lambda}\left{\lambda:|\mathbf{x}-\mathbf{f}(\lambda)|=\inf { \mu}|\mathbf{x}-\mathbf{f}(\mu)|\right}

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## MATLAB代写

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