### 机器学习代写|流形学习代写manifold data learning代考|ICML2022

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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|History of Probabilistic Dimensionality Reduction

The probabilistic variants of many of the spectral dimensionality reduction methods were gradually developed and proposed. For example, probabilistic $P C A[50,62]$ is the stochastic version of PCA, and it assumes that data are obtained by the addition of noise to the linear projection of a latent random variable. It can be demonstrated that PCA is a special case of probabilistic PCA, where the variance of noise tends to zero. The probabilistic PCA, itself, is a special case of factor analysis [17]. In the factor analysis, the different dimensions of noise can be correlated, while they are uncorrelated and isometric in the probabilistic PCA. The factor analysis and the probabilistic PCA use expectation maximization and variational inference on the data.

The linear spectral dimensionality reduction methods, such as PCA and FDA, learn a projection matrix from the data for better data representation or more separation between classes. In 1984, it was surprisingly determined that if a random matrix is used for the linear projection of the data, without being learned from the data, it represents data well! The correctness of this mystery of random projection was proven by the Johnson-Lindenstrauss lemma [33], which put a bound on the error of preservation of the distances in the subspace. Later, nonlinear variants of random projection were developed, including random Fourier features [48] and random kitchen sinks [49].

Sufficient Dimension Reduction $(S D R)$ is another family of probabilistic methods, whose first method was proposed in 1991 [39]. It is used for finding a transformation of data to a lower-dimensional space, which does not change the conditional of labels given data. Therefore, the subspace is sufficient for predicting labels from projected data onto the subspace. SDR was mainly proposed for high-dimensional regression, where the regression labels are used. Later, Kernel Dimensionality Reduction $(K D R)$ was proposed [18] as a method in the family of SDR for dimensionality reduction in machine learning.

Stochastic Neighbour Embedding (SNE) was proposed in 2003 [30] and took a probabilistic approach to dimensionality reduction. It attempted to preserve the probability of a point being a neighbour of others in the subspace. A problem with SNE was that it could not find an optimal subspace because it was not possible for it to preserve all the important information of high-dimensional data in the low-dimensional subspace. Therefore, t-SNE was proposed [41], which used another distribution with more capacity in the subspace. This allowed t-SNE to preserve a larger amount of information in the low-dimensional subspace. A recent successful probabilistic dimensionality reduction method is the Uniform Manifold Approximation and Projection (UMAP) [43], which is widely used for data visualization. Today, both t-SNE and UMAP are used for high-dimensional data visualization, especially in the visualization of extracted features in deep learning. They have also been widely used for visualizing high-dimensional genome data.

## 机器学习代写|流形学习代写manifold data learning代考|History of Neural Network-Based Dimensionality

Neural networks are machine learning models modeled after the neural structure of the human brain. Neural networks are currently powerful tools for representation learning and dimensionality reduction. In the 1990s, researchers’ interest in neural networks decreased; this was called the winter of neural networks. This winter occurred mainly because networks could not become deep, as gradients vanished after many layers of network during optimization. The success of kernel support vector machines [10] also exaggerated this winter. In 2006, Hinton and Salakhutdinov demonstrated that a network’s weights can be initialized using energy-based training, where the layers of the network are considered stacks of Restricted Boltzmann Machines (RBM) [1,31]. RBM is a two-layer structure of neurons, whose weights between the two layers are trained using maximum likelihood estimation [68]. This initialization saved the neural network from the vanishing gradient problem and ended the neural networks’ winter. A deep network using RBM training was named the deep belief network [29]. Although, later, the proposal of the ReLU activation function [23] and the dropout technique [59] made it possible to train deep neural networks with random initial weights [24].

In fundamental machine learning, people often extract features using traditional dimensionality reduction and then apply the classification, regression, or clustering task afterwards. However, modern deep learning extracts features and learns embedding spaces in the layers of the network; this process is called end-to-end. Therefore, deep learning can be seen as performing a form of dimensionality reduction as part of its model. One problem with end-to-end models is that they are harder to troubleshoot if the performance is not satisfactory on a part of the data. The insights and meaning of the data coming from representation learning are critical to fully understand a model’s performance. Some of these insights can be useful for improving or understanding how deep neural networks operate. Researchers often visualize the extracted features of a neural network to interpret and analyze why deep learning is working properly on their data.

Deep metric learning [35] utilizes deep neural networks for extracting lowdimensional descriptive features from data at the last or one-to-last layer of the network. Siamese networks [11] are important network structures for deep metric learning. They contain several identical networks that share their weights, but have different inputs. Contrastive loss [27] and triplet loss [56] are two well-known loss functions that were proposed for training Siamese networks. Deep reconstruction autoencoders also make it possible to capture informative features at the bottleneck between the encoder and decoder.

## 机器学习代写|流形学习代写manifold data learning代考|History of Probabilistic Dimensionality Reduction

PCA 和 FDA 等线性光谱降维方法从数据中学习投影矩阵，以实现更好的数据表示或更好的类间分离。1984年，令人惊奇地确定，如果用一个随机矩阵来做数据的线性投影，不用从数据中学习，就可以很好地表示数据！Johnson-Lindenstrauss 引理 [33] 证明了这种随机投影之谜的正确性，该引理限制了子空间中距离保存的误差。后来，开发了随机投影的非线性变体，包括随机傅立叶特征 [48] 和随机厨房水槽 [49]。

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。