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I was wondering whether it makes sense to do PCA after robust PCA. Suppose I have a matrix $X$ and if I do robust PCA I would get:

$$X=A+E$$

And if I do PCA over $A$ would this make sense as a preprocessing step? The matrix $X$ is highly sparse (less than 10% is populated, with a size of 15K by 20K).

An introduction to robust PCA, if more specific details are needed: I would like to use the inexact ALM method to do this.

My objective is to reduce the dimensionality of $X$, but it contains some extreme outliers.

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    $\begingroup$ Robust PCA of Candes et al. decomposes matrix $X$ into matrix $A$ that has low rank (and is not sparse) and a matrix $E$ that is sparse (and is not low rank). So $A$ is supposed to be low rank. What's the point of another PCA step then? You did not specify what is the goal of your analysis; what is it supposed to be a preprocessing step for? Apart from that, I am not sure that robust PCA makes sense when $X$ is sparse; it is based on the assumption that $A$ is non-sparse and low rank, hence it seems to me that $X$ is supposed to be a dense matrix for robust PCA to make sense. $\endgroup$ – amoeba Aug 13 '15 at 13:26
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Yes, it makes sense.

Classical PCA is highly sensitive to outliers. By using robust PCA, you can identify these outliers and get rid of them.

About Classical PCA: - PCA is sensitive to outliers - PCA does not require distributional assumptions - PCA only considers orthogonal transformations (rotations) of the original variables) - Dimension reduction can only be achieved if the original variables were correlated. Otherwise, PCA is only re-ordering them occording to their variance!!! - PCA is not scale invariant. Often (but not always) scaling is preferred.

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