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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|Laplacian of Graph Sum

Here we are primarily interested in knowing how to derive Laplacian of a resultant graph derived from two different graphs of the same order (on a given data set). In fact we are superimposing two graphs having the same set of vertices together to form a new graph. We do graph fusion as we are interested in combining a local-neighborhood graph and a global graph which we will describe later on.

Harary in [7] introduced a graph operator called Graph Sum; the operator is denoted by $\oplus: G \times H \rightarrow J$, to sum up two graphs of the same order $(|V(G)|=|V(H)|=|V(J)|)$. The operator is quite simple – adjacency matrices of each graph are numerically added to form the adjacency matrix of the summed graph. Let $G$ and $H$ be two graphs of order $n$, and the new summed graph be denoted by $J$ as shown in Figure 2.1. Furthermore, let $A_{G}$, $A_{H}$, and $A_{J}$ be the adjacency of each graph respectively. Then
$$J=G \oplus H,$$

Figure 2.1: The top-left figure shows a graph $G$; top-right figure shows an MST graph $H$; and the bottom-left figure shows the graph sum $J=G \oplus H$. Note how the graphs superimpose on each other to form a new graph.
and
$$A_{J}=A_{G}+A_{H} .$$
From Definition 2, it is obvious that
$$L_{w}(J)=L_{w}(G)+L_{w}(H) .$$

## 机器学习代写|流形学习代写manifold data learning代考|Global Information of Manifold

Global information has not been used in manifold learning since it is widely believed that global information may capture unnecessary data (like ambient data points) that should be avoided when dealing with manifolds.

However, some recent research results show that that it might be useful to to explore global information in a more constrained manner for manifold learning. Costa and Hero show that it is possible to use a Geodesic Minimum Spanning Tree (GMST) on the manifold to estimate the intrinsic dimension and intrinsic entropy of the manifold [4].

Costa and Hero showed in the following theorem that is possible to learn the intrinsic entropy and intrinsic dimension of a non-linear manifold by extending the BHH theorem [1], a well-known result in Geometric Probability.

Theorem: [Generalization of BHH Theorem to Embedded manifolds: [4]] Let $\mathcal{M}$ be a smooth compact m-dimensional manifold embedded in $\mathbb{R}^{d}$ through the diffeomorphism $\phi: \Omega \rightarrow \mathcal{M}$, and $\Omega \in \mathbb{R}^{d}$. Assume $2 \leq m \leq d$ and $0<\gamma<m$. Suppose that $Y_{1}, Y_{2}, \ldots$ are iid random vectors on $\mathcal{M}$ having a common density function $f$ with respect to a Lebesgue measure $\mu_{\mathcal{M}}$ on $\mathcal{M}$. Then the length functional $T_{\gamma}^{\mathbb{R}^{m}} \phi_{-1}\left(Y_{n}\right)$ of the MST spanning $\phi^{-1}\left(Y_{n}\right)$ satisfies the equation shown below in an almost sure sense:

$$\lim {n \rightarrow \infty} \frac{T{\gamma}^{\mathbb{R}^{m}} \phi_{-1}\left(Y_{n}\right)}{n^{\frac{(d-1)}{d}}}=$$
where $\alpha=(m-\gamma) / m$, and is always between $0<\alpha<1, J$ is the Jacobian, and $\beta_{m}$ is a constant which depends on $m$.

Based on the above theorem we use MST on the entire data set as a source of global information. For more details see [4], and more background information see [15] and [13].
The basic principle of GLEM is quite straightforward. The objective function that is to be minimized is given by the following (it is has the same flavor and notation used in [2]):
\begin{aligned} & \sum_{i, j}\left|\mathbf{y}^{(\mathbf{i})}-\mathbf{y}^{(\mathbf{j})}\right|_{2}^{2}\left(W_{i j}^{N N}+W_{i j}^{M S T}\right) \ =& \operatorname{tr}\left(\mathbf{Y}^{T} L\left(G_{N N}\right) \mathbf{Y}+\mathbf{Y}^{T} L\left(G_{M S T}\right) \mathbf{Y}\right) \ =& \operatorname{tr}\left(\mathbf{Y}^{T}\left(L\left(G_{N N}\right)+L\left(G_{M S T}\right)\right) \mathbf{Y}\right) \ =& \operatorname{tr}\left(\mathbf{Y}^{T} L(J) \mathbf{Y}\right) \end{aligned}

# 流形学习代写

## 机器学习代写|流形学习代写manifold data learning代考|Laplacian of Graph Sum

Harary 在 [7] 中引入了一个称为 Graph Sum 的图算子；运算符表示为 $\oplus: G \times H \rightarrow J$, 总结两个相同顺序的图 $(|V(G)|=|V(H)|=|V(J)|)$. 运算符非常简单 将每个图的邻接矩阵数字相加，形成求和图的邻接矩阵。让 $G$ 和 $H$ 是两个顺序图 $n$ ，新的求和图表示为 $J$ 如图 2.1 所示。此外，让 $A_{G}, A_{H}$ ，和 $A_{J}$ 分别为每个 图的邻接关系。然后
$$J=G \oplus H,$$

$$A_{J}=A_{G}+A_{H} .$$

$$L_{w}(J)=L_{w}(G)+L_{w}(H) .$$

## 机器学习代写|流形学习代写manifold data learning代考|Global Information of Manifold

Costa 和 Hero 在以下定理中表明，可以通过扩展 BHH 定理 [1] 来学习非线性流形的内在樀和内在维数，这是几何概率中的一个众所周知的结果。

$$\lim n \rightarrow \infty \frac{T \gamma^{\mathbb{R}^{m}} \phi_{-1}\left(Y_{n}\right)}{n^{\frac{(d 1)}{d}}}=$$

$$\sum_{i, j}\left|\mathbf{y}^{(\mathbf{i})}-\mathbf{y}^{(\mathbf{j})}\right|{2}^{2}\left(W{i j}^{N N}+W_{i j}^{M S T}\right)=\operatorname{tr}\left(\mathbf{Y}^{T} L\left(G_{N N}\right) \mathbf{Y}+\mathbf{Y}^{T} L\left(G_{M S T}\right) \mathbf{Y}\right)=\operatorname{tr}\left(\mathbf{Y}^{T}\left(L\left(G_{N N}\right)+L\left(G_{M S T}\right)\right) \mathbf{Y}\right)=\quad \operatorname{tr}\left(\mathbf{Y}^{T} L(J) \mathbf{Y}\right)$$

## 有限元方法代写

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## MATLAB代写

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