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

Figures $2.5-2.7$ show the results after using LEM for different values of $k$. As the value of $k$ increases from 1 to higher values we notice the spreading of the embedded data. The bottom subplot shows the nearest neighbor graph with $k=1$ as shown in Figure 2.7. The right plot shows the embedding of the graph. It is interesting to observe how the embedded data loses its local neighborhood information. The embedding practically happens along the second principal eigenvector (the first being Zero Vector). As the value of $k$ is increased to 2, we observe that embedding happens along the second and third principal axes. See Figure 2.7. For $k=1$ the graph is highly disconnected and for $k=2$ the graphs has much less isolated pieces of graphs. One interesting thing to observe is that as the connectivity of the graph increases the low-dimensional representation begins to preserve the local information.
The graph with $k=2$ and its embedding is shown in Figure 2.8. Increasing the neighborhood information to 2 neighbors is still not able to represent the continuity of the original manifold. Figure $2.7$ shows the graph with $k=3$ and its embedding. Increasing the neighborhood information to 3 neighbors better represents the continuity of the original manifold. Figure $2.5$ shows the graph with $k=5$ and its embedding. Increasing the neighborhood information to 5 neighbors better represents the continuity of the original manifold. Similar results are obtained by increasing the the number of neighbors, however, it should be noted that when the number of neighbors is very high then the graph starts to get influenced by ambient neigbhors.

We see similar results for the face images. The three plots in Figure $2.6$ show the embedding results obtained using LEM when the neighborhood graphs are created using $k=1, k=2$, and $k=5$. The top and the middle plot validate the limitation of LEM.

## 机器学习代写|流形学习代写manifold data learning代考|Bibliographical and Historical Remarks

Dimensionality reduction is an important research area in data analysis with an extensive research literature. Both linear and non-linear methods exist, and each category has both supervised and unsupervised versions. In this section we will briefly mention some of the salient works that have been proposed in the area of locally preserving manifold learning: see $[8]$ for a broader survey.

Lee and Seung [12] showed that many high dimensional data such as a series of related images, video frames, etc. lie on a much lower-dimensional manifold instead of being scattered throughout the feature space. This particular observation has motivated researchers to develop dimension reduction algorithms that try to learn an embedded manifold in a high-dimensional space.

ISOMAP [14] learns the manifold by exploring geodesic distances. In fact the algorithm tries to preserve the geometry of the data on the manifold by noting the points in the neighborhood of each point. The algorithm is defined as such:

1. Form a neighborhood graph $G$ for the dataset, based, for instance, on the $K$ nearest neighbors of each point $x_{i}$ –
2. For every pair of nodes in the graph, compute the shortest path, using Dijkstras algorithm, as an estimate of intrinsic distance on the data manifold. The weights of edges of the graphs are computed based on the Euclidean distance measure.
3. Classical Multi-Dimensional Scaling algorithm is computed using these pairwise distances to find a lower dimensional embedding $y_{i}$.

Bernstein et al. [22] have described the convergence properties of the estimation procedure for the intrinsic distances. For large and dense data sets, computation of pairwise distances is time consuming, and moreover the calculation of eigenvalues can be computationally intensive for large data sets. Such constraints have motivated researchers to find simpler variations of the Isomap algorithm. One such algorithm uses subsampled data called landmarks. Firstly, it calculates Isomap for random points called landmarks and between those landmarks a simple triangulation algorithm is applied.

Locally Linear Embedding (LLE) is an unsupervised learning method based on global and local optimization [11]. It is is similar to Isomap in the sense that it generates a graphical representation of the data set. However, it is different from Isomap as it only attempts to preserve local structures of the data. Because of the locality property used in LLE, the algorithm allows for successful embedding of nonconvex manifolds. An important point to be noted is that LLE creates the local properties of a manifold using the linear combinations of $k$ nearest neighbors of the data $x_{i}$. LLE attempts to create a local regression like model and thereby tries to fit a hyperplane through the data point $x_{i}$. This appears to be reasonable for smooth manifolds where the nearest neighbors align themselves well in a linear space. For very non-smooth or noisy data sets, LLE does not perform well. It has been noted that LLE preserves the reconstruction weights in the space of lower dimensionality, as the reconstruction weights of a data point are invariant to linear transformational operations like translation, rotation, etc.

## 机器学习代写|流形学习代写manifold data learning代考|Arkadas Ozakin, Nikolaos Vasiloglou II, Alexander Gray

Much of the recent work in manifold learning and nonlinear dimensionality reduction focuses on distance-based methods, i.e., methods that aim to preserve the local or global (geodesic) distances between data points on a submanifold of Euclidean space. While this is a promising approach when the data manifold is known to have no intrinsic curvature (which is the case for common examples such as the “Swiss roll”), classical results in Riemannian geometry show that it is impossible to map a $d$-dimensional data manifold with intrinsic curvature into $\mathbb{R}^{d}$ in a manner that preserves distances. Consequently, distance-based methods of dimensionality reduction distort intrinsically curved data spaces, and they often do so in unpredictable ways. In this chapter, we discuss an alternative paradigm of manifold learning. We show that it is possible to perform nonlinear dimensionality reduction by preserving the underlying density of the data, for a much larger class of data manifolds than intrinsically flat ones, and demonstrate a proof-of-concept algorithm demonstrating the promise of this approach.

Visual inspection of data after dimensional reduction to two or three dimensions is among the most common uses of manifold learning and nonlinear dimensionality reduction. Typically, what is sought by the user’s eye in two or three-dimensional plots is clustering and other relationships in the data. Knowledge of the density, in principle, allows one to identify such basic structures as clusters and outliers, and even define nonparametric classifiers; the underlying density of a data set is arguably one of the most fundamental statistical objects that describe it. Thus, a method of dimensionality reduction that is guaranteed to preserve densities may well be preferable to methods that aim to preserve distances, but end up distorting them in uncontrolled ways.

Many of the manifold learning methods require the user to set a neighborhood radius $h$, or, for $k$-nearest neighbor approaches, a positive integer $k$, to be used in determining the neighborhood graph. Most of the time, there is no automatic way to pick the appropriate values of the tweak parameters $h$ and $k$, and one resorts to trial and error, looking for values that result in reasonable-looking plots. Kernel density estimation, one of the most popular and useful methods of estimating the underlying density of a data set, comes with a natural way to choose $h$ or $k$; it suggests to us to pick the value that maximizes a cross-validation score for the density estimate. While the usual kernel density estimation does not allow one to estimate the density of data on submanifolds of Euclidean space, a small modification

allows one to do so. This modification and its ramifications are discussed below in the context of density-preserving maps.

The chapter is organized as follows. In Section 3.2, using a theorem of Moser, we prove the existence of density preserving maps into $\mathbb{R}^{d}$ for a large class of $d$-dimensional manifolds, and give an intuitive discussion on the nonuniqueness of such maps. In Section 3.3, we describe a method for estimating the underlying density of a data set on a Riemannian submanifold of Euclidean space. We state the main result on the consistency of this submanifold density estimator, and give a bound on its convergence rate, showing that the latter is determined by the intrinsic dimensionality of the data instead of the full dimensionality of the feature space. This, incidentally, shows that the curse of dimensionality in the widely-used method of kernel density estimation is not as severe as is generally believed, if the method is properly modified for data on submanifolds. In Section 3.4, using a modified version of the estimator defined in Section 3.3, we describe a proof-of-concept algorithm for density preserving maps based on semidefinite programming, and give experimental results. Finally, in Sections $7.7$ and 3.6, we summarize the chapter and discuss relevant bibliography.

## 机器学习代写|流形学习代写manifold data learning代考|Bibliographical and Historical Remarks

Lee 和 Seung [12] 表明，许多高维数据（例如一系列相关图像、视频帧等）位于低得多的流形上，而不是分散在整个特征空间中。这一特殊观察促使研究人员开发降维算法，试图在高维空间中学习嵌入式流形。

ISOMAP [14] 通过探索测地距离来学习流形。事实上，该算法试图通过注意每个点附近的点来保留流形上数据的几何形状。算法定义如下：

1. 形成邻域图G对于数据集，例如，基于ķ每个点的最近邻X一世 –
2. 对于图中的每一对节点，使用 Dijkstras 算法计算最短路径，作为数据流形上内在距离的估计。图的边权重是根据欧几里得距离度量计算的。
3. 使用这些成对距离计算经典的多维缩放算法以找到较低维的嵌入是一世.

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## MATLAB代写

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