I have a set of files consisting of randomly selected points from a dataset, each file belonging to a particular class. Each row in these files contains the coordinates in n-space of the point. I'd like to compare the distributions in n-space of each of these files - and am inspired by the K-S test for comparing histograms. From what I've read this method doesn't extend well to multivariate data. I had previously used PCA - but all of my variance collapsed into a single noisy dimension and clustering methods were useless.
My question - is there a reason I shouldn't just use an average of the K-S values across the histogram for each of the n-dimensions as a metric for the goodness of fit? Is there a better method for comparing these distributions?