Yes. Been there. Done that. The correlation is stunningly good.
On Fri, Jul 1, 2011 at 4:22 PM, Lance Norskog <[email protected]> wrote: > 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] >
