Continuous Latent Variables

Principal Component Analysis

Introduction

Principal Component Analysis is widely used for applications such as dimensionality reduction,lossy data compression,feature extraction,and data visualization.Also known as the Karhunen-Loeve transform.There are two definitions giving rise to the same algorithm.PCA can be defined as the orthogonal projection of the data onto a lower dimensional linear space,known as the principal subspace,such that the variance of the projected data is maximized.Equivalently,it can be defined as the linear projection the minimizes the average projection cost, defined as the linear projection that minimizes the average projection cost,defined as the mean squared distance between the data points and their projections.

Maximum variance formulation

Consider a data set of observations $\{x_n\}$ where $n = 1,...,N$,and $x_n$ is a Euclidean variable with dimensionality D. Our goal is to project the data onto a space having dimensionality $M < D$ while maximizing the variance of the projected data.We define the direction of this space using a D-dimensional unit vector $\mathbf{u_1^T}\mathbf{u_1} = 1$.Each data point $\mathbf{x_n}$ is then projected onto a scalar value $\mathbf{u_1^T}\mathbf{x_n}$.The mean of the projected data is $\mathbf{u_1^T}\bar{\mathbf{x}}$ where the $\bar{\mathbf{x}}$ is the sample set mean given by $$\begin{aligned} \bar{\mathbf{x}} = \frac{1}{N}\sum_{n=1}^{N}{\mathbf{x_n}}\end{aligned}$$ and the variance of the projected data is given by $$\begin{aligned} \frac{1}{N}\sum_{n=1}^{N}\{\mathbf{u_1^T}\mathbf{x_n} - \mathbf{u_1^T}\bar{\mathbf{x}}\}^2 &= \frac{1}{N}\sum_{n=1}^{N}{\{\mathbf{u_1^T}(\mathbf{x_n} - \bar{\mathbf{x}})\}^2} \\ &= \frac{1}{N}\sum_{n=1}^{N}{\{\mathbf{u_1^T(\mathbf{x_n - \bar{\mathbf{x}}})(\mathbf{x_n -\bar{x}})^T\mathbf{u_1^T} } \}} \\ &= \mathbf{u_1^T}\mathbf{S}\mathbf{u_1}\end{aligned}$$ where $\mathbf{S}$ is the data covariance matrix defined by $$\begin{aligned} \mathbf{S} = \frac{1}{N}\sum_{n=1}^{N}(\mathbf{x_n}-\bar{\mathbf{x}})(\mathbf{x_n}-\mathbf{\bar{x}})^T\end{aligned}$$ We now maximize the projected variance $\mathbf{u_1^T}\mathbf{S}\mathbf{u_1}$ with respect to $\mathbf{u_1}$,which is a constrained maximization to prevent $\parallel\mathbf{u_1}\parallel\rightarrow \infty$ .The appropriate constraint comes from the normalization condition $\mathbf{u_1^T}\mathbf{u_1}=1$.To enforce this constraint,we introduce a Lagrange multiplier that we shall denote by $\lambda_1$,and then make an unconstrained maximization of $$\mathbf{u_1^T}\mathbf{S}\mathbf{u_1} + \lambda_1(1-\mathbf{u_1^T}\mathbf{u_1})$$ By setting the derivative with respect to $\mathbf{u_1}$ equal to zero,we see that this quantity will have a stationary point when $$\mathbf{S}\mathbf{u_1} = \lambda_1\mathbf{u_1}$$ which says that $\mathbf{u_1}$ must be an eigenvector of $\mathbf{S}$.If we left-multiply by $\mathbf{u_1^T}$ and make use of $\mathbf{u_1^T}{u_1} = 1$,we see that the variance is given by $$\mathbf{u_1^TSu_1} = \lambda_1$$ and so the variance will be a maximum when we set $\mathbf{u_1}$ equal to the eigenvector having the largest eigenvalue $\lambda_1$.This eigenvector is known as the first principal component.

Minimum-error formulation

Applications of PCA

PCA for high-dimensional data

Probabilistic PCA

Kernel PCA

Nonlinear Latent Variable Models