How to use reduced dimensions (of a PCA) for detection purposes? A general question aiming at the application of a PCA: I want to detect abnormal data points and therefore I want to use a PCA for it at first. The next step is to try several distance functions or quantils of distributions or something similar to check if it is working.
My problem is: How do I know this when I operate on the dimensionally reduced data set? Or in other words: How do I transform it back to see if the certain marked data points match the ones from the original non-transformed data?

I expect something to do like here: 



How do I know that this outlier in the transformed/reduced data is sensefull? I think I have to cross-check it somehow in the original data?
Well, mathematically speaking, applying PCA on a bunch of data points usually means (there are some variants) removing the mean and rotate to get uncorrelated features. Then you reduce dimensionality by discarding some of the uncorrelated features.

You may get the corresponding points in the original feature space by doing the inverse rotation and adding the mean back again. Of course, this points now will be contained in a shape of less dimension than before. For example, if you went from 3 original features to 2 uncorrelated features, then perhaps your original data points had some 3D shape, say a football. Then your new points (transformed-inverse transformed) will have the shape of an ellipse.