After running PCA on my data set, I noticed that using the three first eigenvectors, a separation between two different classes is still achievable (doing PCA on data from two classes). Unfortunately, it doesn't scale up very well and when I feed in data from 15 different classes, I get a big blob of dots when I plot the reprojected data in three dimensions. I want to find out if those classes are actually separated in higher dimensions. Is there techniques used to visualize scatter plots of data in more than 3 dimensions?
I have read about using different shapes (dots, triangles, squares, and their respective area) to fit more information in a 3D scatter plot but how would you code that relative to the values delivered by PCA?