How many dimensions to reduce to when doing PCA?
After performing the PCA algorithm you get the principal components, sorted by the amount of information they hold. If you keep the whole set there is no information lost. Removing them one by one and projecting them back onto the original space you can calculate the information loss. You can plot this information loss against number of principal components removed and see if it makes an 'elbow' where it makes sense. A lot of this depends on your use case though.