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I am not very familiar with LDA but was interested in using it to find multiple, orthogonal discriminant dimensions for data with 2 class labels. I understand that standard that the point of LDA is to find a single axis that discriminates 2 classes so it will necessarily find a single discriminant dimension but I was wondering whether, having found this axis, I would be able to factor it out of the data

i.e. say $\beta$ $\in \mathbb{R}^{m} $are the LDA coefficients that project the data $D \in \mathbb{R}^{n x m}$ where $n$ is the number of samples and $m$ is the number of features of each sample, onto the discriminant axis. Then calculate

$$D2 = D - \left( \frac{\beta\cdot(\beta^T\cdot D^T)}{(\beta^T\cdot\beta)} \right) ^T $$

and rerun LDA on $D2$ again to find a second orthogonal discriminant dimension. I am interested in using this approach mainly for data visualisation but am not sure whether it make sense? If so is there a name for this method?

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Finding the discriminant directions is equivalent to finding the eigen values of the matrix $S_w^{-1} S_b$, where $S_w$ is the within scatter matrix and $S_b$ the between scatter matrix. For $C$ classes, this matrix has rank $C-1$, which means that you can only find $C-1$ discriminant directions.

In your case of only two classes, you cannot find more than one direction.

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