Is applying dimension reduction to mixed type data valid for outlier detection after that? I'm facing with anomaly detection (outlier detection) task with mixed (numerical and categorical) multi-feature data set. I understand that many of the possible multivariate outlier detection methods require numerical only features so I'm wondering to apply mixed-type dimension reduction technique (like FAMD) to obtain full-numerical features.
My question is if is it a valid approach? I somewhat fear that the very outlier nature of outliers may be neutralized by the dimension reduction's effect.

It sounds like an interesting idea to use FAMD to obtain purely numerical data. However, as you have already pointed out, dimension reduction would severely interfere with the anomaly detection task, since isolation information could easily be removed. Think of the popular simple outlier detection method using PCA: There, you use the distance of a point to the principal component subspace as the outlier score, but, when using this very PCA for dimensionality reduction, those outlier scores would all become zero.
The approach I would suggest is to use one-hot encoding for the categorical variables and then to apply Isolation Forest or variants thereof like e.g. Extended Isolation Forest. This method is very well established and known to work well.
