### 机器学习代写|流形学习代写manifold data learning代考|Robust Laplacian Eigenmaps Using Global Information

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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|Shounak Roychowdhury and Joydeep Ghosh

Dimensionality reduction is an important process that is often required to understand the data in a more tractable and humanly comprehensible way. This process has been extensively studied in terms of linear methods such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), Factor Analysis etc. [8]. However, it has been noticed that many high dimensional data, such as a series of related images, lie on a manifold $[12]$ and are not scattered throughout the feature space.

Belkin and Niyogi in [2] proposed Laplacian Eigenmaps (LEM), a method that approximates the Laplace-Beltrami Operator which is able to capture the properties of any Riemaniann manifold. The motivation of our work derives from our experimental observations that when the graph that used Laplacian Eigenmaps (LEM) [2] is not well-constructed (either it has lot of isolated vertices or there are islands of subgraphs) the data is difficult to interpret after a dimension reduction. This paper discusses how global information can be used in addition to local information in the framework of Laplacian Eigenmaps to address such situations. We make use of an interesting result by Costa and Hero that shows that Minimum Spanning Tree on a manifold can reveal its intrinsic dimension and entropy [4]. In other words, it implies that MSTs can capture the underlying global structure of the manifold if it exists. We use this finding to extend the dimension reduction technique using LEM to exploit both local and global information.

LEM depends on the Graph Laplacian matrix and so does our work. Fiedler initially proposed the Graph Laplacian matrix as a means to comprehend the notion of algebraic connectivity of a graph [6]. Merris has extensively discussed the wide variety of properties of the Laplacian matrix of a graph such as invariance, on various bounds and inequalities, extremal examples and constructions, etc., in his survey [10]. A broader role of the Laplacian matrix can be seen in Chung’s book on Spectral Graph Theory [3].

The second section touches on the Graph Laplacian matrix. The role of global information in manifold learning is then presented, followed by our proposed approach of augmenting LEM by including global information about the data. Experimental results confirm that global information can indeed help when the local information is limited for manifold learning.

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

Let us consider a weighted graph $G=(V, E)$, where $V=V(G)=\left{v_{1}, v_{2}, \ldots, v_{n}\right}$ is the set of vertices (also called vertex set) and $E=E(G)=\left{e_{1}, e_{2}, \ldots, e_{n}\right}$ is the set of edges (also called edge set). The weight $w$ function is defined as $w: V \times V \rightarrow \Re$ such that $w\left(v_{i}, v_{j}\right)=w\left(v_{j}, v_{i}\right)=w_{i j}$.
Definition 1: The Laplacian [6] of a graph without loops of multiple edges is defined as the following:
$$L(G)= \begin{cases}d_{v_{i}} & \text { if } v_{i}=v_{j} \ -1 & \text { if } v_{i} \text { are } v_{j} \text { adjacent } \ 0 & \text { Otherwise }\end{cases}$$
Fiedler [6] defined the Laplacian of a graph as a symmetric matrix for a regular graph, where $A$ is an adjacency matrix ( $A^{T}$ is the transpose of adjacency matrix), $I$ is the identity matrix, and $n$ is the degree of the regular graph:
$$L(G)=n I-A .$$
A definition by Chung (see [3]) – which is given below – generalizes the Laplacian by adding the weights on the edges of the graph. It can be viewed as Weighed Graph Laplacian. Simply, it is a difference between the diagonal matrix $D$ and $W$, the weighted adjacency matrix.
$$L_{W}(G)=D-W,$$
where the diagonal element in $D$ is defined as $d_{v_{i}}=\sum_{j=1}^{n} w\left(v_{i}, v_{j}\right)$.
Definition 2: The Laplacian of weighted graph (operator) is defined as the following:
$$L_{w}(G)= \begin{cases}d_{v_{i}}-w\left(v_{i}, v_{j}\right) & \text { if } v_{i}=v_{j} \ -w\left(v_{i}, v_{j}\right) & \text { if } v_{i} \text { are } v_{j} \text { connected } \ 0 & \text { otherwise. }\end{cases}$$
$L_{w}(G)$ reduces to $L(G)$ when the edges have unit weights.

## 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:

$$\lim {n \rightarrow \infty} \frac{T{\gamma}^{2^{m}} \phi_{-1}\left(Y_{n}\right)}{n \frac{(d-1)}{d}}=$$
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}
where $\mathbf{y}^{(i)}=\left[y_{1}(i), \ldots, y_{m}(i)\right]^{T}$, and $m$ is the dimension of embedding. $W_{i j}^{N N}$ and $W_{i j}^{M S T}$ are weighted matrices of k-Nearest Neighbor graph and the MST graph respectively. In other words, we have
$$\operatorname{argmin}{\mathbf{Y T} \mathbf{Y}=\mathbf{I}} \mathbf{Y}^{T} L \mathbf{Y}$$ such that $Y=\left[\mathbf{y}{\mathbf{1}}, \mathbf{y}{\mathbf{2}}, \ldots, \mathbf{y}{\mathbf{m}}\right]$ and $\mathbf{y}^{(\mathbf{i})}$ is the $m$-dimensional representation of $i^{\text {th }}$ vertex. The solutions to this optimization problem are the eigenvectors of the generalized eigenvalue problem
$$L \mathbf{Y}=\Lambda D \mathbf{Y}$$
The GLEM algorithm is described in Algorithm $1 .$

## 机器学习代写|流形学习代写manifold data learning代考|Shounak Roychowdhury and Joydeep Ghosh

Belkin 和 Niyogi 在 [2] 中提出了 Laplacian Eigenmaps (LEM)，这是一种近似 Laplace-Beltrami 算子的方法，能够捕获任何黎曼流形的属性。我们工作的动机源于我们的实验观察，即当使用拉普拉斯特征图 (LEM) [2] 的图构造不完善（它有很多孤立的顶点或存在子图岛）时，数据很难解释降维后。本文讨论了如何在拉普拉斯特征图框架中使用全局信息和局部信息来解决这种情况。我们利用了 Costa 和 Hero 的一个有趣结果，该结果表明流形上的最小生成树可以揭示其内在维度和熵 [4]。换句话说，这意味着 MST 可以捕获流形的潜在全局结构（如果存在）。我们利用这一发现来扩展使用 LEM 的降维技术，以利用本地和全局信息。

LEM 依赖于图拉普拉斯矩阵，我们的工作也是如此。Fiedler 最初提出图拉普拉斯矩阵作为理解图的代数连通性概念的一种手段 [6]。Merris 在他的调查 [10] 中广泛讨论了图的拉普拉斯矩阵的各种属性，例如不变性、各种边界和不等式、极值示例和构造等。拉普拉斯矩阵的更广泛作用可以在 Chung 的关于谱图理论的书中看到 [3]。

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

Fiedler [6] 将图的拉普拉斯算子定义为正则图的对称矩阵，其中一种是一个邻接矩阵 (一种吨是邻接矩阵的转置），一世是单位矩阵，并且n是正则图的度数：

Chung 的定义（见 [3]）——下面给出——通过在图的边缘上添加权重来概括拉普拉斯算子。它可以看作是加权图拉普拉斯算子。简单来说，就是对角矩阵的区别D和在，加权邻接矩阵。

## 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熵和内在维数，这是几何概率中的一个众所周知的结果。

GLEM 的基本原理非常简单。要最小化的目标函数由以下给出（它与 [2] 中使用的风格和符号相同）：
∑一世,j|是(一世)−是(j)|22(在一世jññ+在一世j米小号吨) =tr⁡(是吨大号(Gññ)是+是吨大号(G米小号吨)是) =tr⁡(是吨(大号(Gññ)+大号(G米小号吨))是) =tr⁡(是吨大号(Ĵ)是).

GLEM 算法在算法中描述1.

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