Thanks! Is this true? - "Preserving pairwise distances" means the relative distances. So the ratios of new distance:old distance should be similar. The standard deviation of the ratios gives a rough&ready measure of the fidelity of the reduction. The standard deviation of simple RP should be highest, then this RP + orthogonalization, then MDS.
On Fri, Jul 1, 2011 at 11:03 AM, Ted Dunning <[email protected]> wrote: > Lance, > > You would get better results from the random projection if you did the first > part of the stochastic SVD. Basically, you do the random projection: > > Y = A \Omega > > where A is your original data, R is the random matrix and Y is the result. > Y will be tall and skinny. > > Then, find an orthogonal basis Q of Y: > > Q R = Y > > This orthogonal basis will be very close to the orthogonal basis of A. In > fact, there are strong probabilistic guarantees on how good Q is as a basis > of A. Next, you project A using the transpose of Q: > > B = Q' A > > This gives you a short fat matrix that is the projection of A into a lower > dimensional space. Since this is a left projection, it isn't quite what you > want in your work, but it is the standard way to phrase things. The exact > same thing can be done with left random projection: > > Y = \Omega A > L Q = Y > B = A Q' > > In this form, B is tall and skinny as you would like and Q' is essentially > an orthogonal reformulation of of the random projection. This projection is > about as close as you are likely to get to something that exactly preserves > distances. As such, you should be able to use MDS on B to get exactly the > same results as you want. > > Additionally, if you start with the original form and do an SVD of B (which > is fast), you will get a very good approximation of the prominent right > singular vectors of A. IF you do that, the first few of these should be > about as good as MDS for visualization purposes. > > On Fri, Jul 1, 2011 at 2:44 AM, Lance Norskog <[email protected]> wrote: > >> I did some testing and make a lot of pretty charts: >> >> http://ultrawhizbang.blogspot.com/ >> >> If you want to get quick visualizations of your clusters, this is a >> great place to start. >> >> -- >> Lance Norskog >> [email protected] >> > -- Lance Norskog [email protected]
