PCA降维算法讲义ppt slides，pca降维算法，课程资源，pptIntrinsic late

文件名称: PCA降维算法讲义ppt slides

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详细说明：PCA降维算法讲义ppt slides，pca降维算法，课程资源，pptIntrinsic latent dimensions In this dataset, there is only 3 degrees of freedom of variability, corresponding to vertical and horizontal translations, and the rotations 333[3 Each image undergoes a random displacement and rotation within some larger image field The resulting images have 100 X 100=10,000 pixels Generative View Each data example generated by first selecting a point from a distribution in the latent space, then generating a point from the conditional distribution in the input space Simplest latent variable models: Assume Gaussian distribution for both latent and observed variables This leads to probabilistic formulation of the Principal Component Analysis and Factor Analysis We will first look at standard pca, and then consider its probabilistic formation Advantages of probabilistic formulation: use of EM for parameter estimation, mixture of Pcas, bayesian PCa Principal Component Analysis Used for data compression, visualization, feature extraction, dimensionality reduction The goal is find M principal components underlying d-dimensional data select the top M eigenvectors of s(data covariance matrix ur,,uMJ project each input vector x into thIs subspace Full projection into M dimensions Two views/derivations takes form Maximize variance(scatter of green points) x1 ·● Minimize error(red-green distance per data point) Maximum variance formulation Consider a dataset X,,.XNJ, Xn E RD Our goal is to project data onto a space having dimensionality M< d Consider the projection into m=1 dimensional space Define the direction of this space using a D-dimensional unit vector u, so that ui u1=l Objective: maximize the variance of the projected data with respect to u1 ∑{uxn-ux}2=uSu1 m=1 where sample mean and data covariance is given by 1 ∑(xn-x)(xn-x)1 =1 Maximum variance formulation Maximize the variance of the projected data U1X 2 1 =1 Must constrain u1=1. Using Langrage multiplier, maximize Su1+入(1 Setting the derivative with respect to u, to zero Su1=入1 Hence u, must be an eigenvector of s The maximum variance of the projected data is given by Optimal u, is principal component (eigenvector with maximal eigenvalue) Minimum error formulation Introduce a complete orthonormal set of D-dimensional basis vectors u Without loss of generality, we can write D T n2 X飞 Rotation of the coordinate system to a new system defined by u Our goal is to represent data points by the projection into M-dimensional subspace(plus some distortion) Represent m-dim linear subspace by the first m of the basis vectors M xmu;十 =M+1 Minimum error formulation Represent M-dim linear subspace by the first m of the basis vectors xnul;十 bi l ui =1 元=M+1 where Zni depend on the particular data point and b are constants Objective: minimize distortion with respect to u;, Z and b N =1 X U Minimizing with respect to znj bi T Hence, the objective reduces to =1=M+1 i=M+1 Minimum error formulation Minimize distortion with respect to u; constraint minimization problem =1 乙=M+1 The general solution is obtained by choosing u; to be eigenvectors of the covariance matrix Suz=入;u The distortion is then given by: J 元=M+1 The objective is minimized when the remaining D-M components are the eigenvectors of s with lowest eigenvalues>same result We will later see a generalization deep autoencoders

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