Similar to Linear discriminant analysis (LDA), a Generalized Discriminant Analysis (GDA) method project the data in the high-dimensional space to a space of fewer dimensions. 
LDA method does not work for nonlinear problems. By mapping the input space into a high dimensional feature space with linear properties, a Generalized Discriminant Analysis (GDA) can deal with nonlinear discriminant analysis.
The mapping process of GDA is similar to the mapping process in Support Vector Machine (SVM).


University of Michigan - Ann Arbor

- Principal Component Analysis (PCA)
- Linear Discriminant Analysis (LDA)
- Generalized Discriminant Analysis (GDA)
- Singular Value Decomposition (SVD)

Methods of dimensionality reduction

Baudat, G, & Anouar, F. (1/10/2000). Generalized Discriminant Analysis Using a Kernel Approach. Neural Computation, 12(10), 2385–2404. Article, CAMBRIDGE: MIT Press.

Generalized Discriminant Analysis Using a Kernel Approach

Linear discriminant analysis (LDA) is another means of classification similar to logistic regression, but better suited to handle non-binary classifications (i.e. when there are >=3 possible output classes).  

LDA assumes Gaussian distributions for observations in each class, with means being class specific and a covariance common to all classes. Then with the help of Bayes' theorem, the probability for one observation being in each class can be estimated. The class with the highest probability would be the classification result.

In this assumption, the classification is equivalent to maximize the *discriminant function* 
$$
\delta_{k}(x)=x^{T} \boldsymbol{\Sigma}^{-1} \mu_{k}-\frac{1}{2} \mu_{k}^{T} \boldsymbol{\Sigma}^{-1} \mu_{k}+\log \pi_{k}
$$
where ${\mu}_{k}$ is the mean for observations in the $k$th class, ${\pi}_{k}$ is the prior probability that an observation belongs to the $k$th class, and $\Sigma$ is the common covariance matrix. The *discriminant function* is linear to $x$, which is why the method is called *linear discriminant analysis*.


Linear Discriminant Analysis (LDA)

Principal component analysis (PCA) is a method of unsupervised learning techniques that functions as a method for dimension reduction in regression models.  

As explained in the textbook, the objective of PCA is *to find a low-dimensional representation of the observations that explain a good fraction of the variance* (ISLR).


Principal Components Analysis (PCA)

Generalized discriminant analysis (GDA)

Singular value decomposition (SVD) is an unsupervised dimensionality reduction technique used for feature extraction.  The singular value decomposition of an m x n matrix M is a factorization of the form U$$\sum$$M*.  U is an m x m unitary matrix. $$\sum$$ is a m x n diagonal matrix. V is a v x v unitary matrix.
Image link: [www.wikiwand.com/en/Singular_value_decomposition](https://www.wikiwand.com/en/Singular_value_decomposition)
 

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