I think that you can do the covariance using Jakes old outer product trick.
Of course you need to do something clever to deal with mean subtraction. 2011/6/23 <[email protected]> > Yes but a M/R job to create the covariance matrix would be required. With > millions of rows that is, unless I am missing something. > > > Doing PCA on 5000 x 5000 matrix is still an in-memory thing. That's > > only 25M doubles, 200MB of memory. Lots of techniques can run > > on that set. You could do Lanczos with whatever rank you want > > on it (but don't worry about distributed lanczos). > > > > 2011/6/23 <[email protected]> > > > >> I will, if it works I may have to make an m/r job for it. All the data > >> we > >> have will be tall and dense (lets say 5000 columns, with millions of > >> rows). Now doing PCA on that will create a covariance matrix that is > >> square and dense. Thanks again guys. > >> > >> -Trevor > >> > >> > Try the QR trick. It is amazingly effective. > >> > > >> > 2011/6/23 <[email protected]> > >> > > >> >> Alright, thanks guys. > >> >> > >> >> > The cases where Lanczos or the stochastic projection helps are > >> cases > >> >> where > >> >> > you have *many* columns but where the data are sparse. If you have > >> a > >> >> very > >> >> > tall dense matrix, the QR method is to be muchly preferred. > >> >> > > >> >> > 2011/6/23 <[email protected]> > >> >> > > >> >> >> Ok, then what would you think to be the minimum number of columns > >> in > >> >> the > >> >> >> dataset for Lanczos to give a reasonable result? > >> >> >> > >> >> >> Thanks, > >> >> >> -Trevor > >> >> >> > >> >> >> > A gazillion rows of 2-columned data is really much better suited > >> to > >> >> >> doing > >> >> >> > the following: > >> >> >> > > >> >> >> > if each row is of the form [a, b], then compute the matrix > >> >> >> > > >> >> >> > [[a*a, a*b], [a*b, b*b]] > >> >> >> > > >> >> >> > (the outer product of the vector with itself) > >> >> >> > > >> >> >> > Then take the matrix sum of all of these, from each row of your > >> >> input > >> >> >> > matrix. > >> >> >> > > >> >> >> > You'll now have a 2x2 matrix, which you can diagonalize by hand. > >> >> It > >> >> >> will > >> >> >> > give you your eigenvalues, and also the right-singular vectors > >> of > >> >> your > >> >> >> > original matrix. > >> >> >> > > >> >> >> > -jake > >> >> >> > > >> >> >> > 2011/6/23 <[email protected]> > >> >> >> > > >> >> >> >> Yes, exactly why I asked it for only 2 eigenvalues. So what is > >> >> being > >> >> >> >> said, > >> >> >> >> is if I have lets say 50M rows of 2 columned data, Lanczos > >> can't > >> >> do > >> >> >> >> anything with it (assuming it puts the 0 eigenvalue in the mix > >> - > >> >> of > >> >> >> the > >> >> >> >> 2 > >> >> >> >> eigenvectors only 1 is returned because of the 0 eigenvalue > >> taking > >> >> up > >> >> >> a > >> >> >> >> slot)? > >> >> >> >> > >> >> >> >> If the eigenvalue of 0 is invalid, then should it not be > >> filtered > >> >> out > >> >> >> so > >> >> >> >> that it returns "rank" number of eigenvalues that could be > >> valid? > >> >> >> >> > >> >> >> >> -Trevor > >> >> >> >> > >> >> >> >> > Ah, if your matrix only has 2 columns, you can't go to rank > >> 10. > >> >> >> Try > >> >> >> >> on > >> >> >> >> > some slightly less synthetic data of more than rank 10. You > >> >> can't > >> >> >> >> > ask Lanczos for more reduced rank than that of the matrix > >> >> itself. > >> >> >> >> > > >> >> >> >> > -jake > >> >> >> >> > > >> >> >> >> > 2011/6/23 <[email protected]> > >> >> >> >> > > >> >> >> >> >> Alright I can reorder that is easy, just had to verify that > >> the > >> >> >> >> ordering > >> >> >> >> >> was correct. So when I increased the rank of the results I > >> get > >> >> >> >> Lanczos > >> >> >> >> >> bailing out. Which incidentally causes a > >> NullPointerException: > >> >> >> >> >> > >> >> >> >> >> INFO: 9 passes through the corpus so far... > >> >> >> >> >> WARNING: Lanczos parameters out of range: alpha = NaN, beta > >> = > >> >> NaN. > >> >> >> >> >> Bailing out early! > >> >> >> >> >> INFO: Lanczos iteration complete - now to diagonalize the > >> >> >> >> tri-diagonal > >> >> >> >> >> auxiliary matrix. > >> >> >> >> >> Exception in thread "main" java.lang.NullPointerException > >> >> >> >> >> at > >> >> >> >> >> > org.apache.mahout.math.DenseVector.assign(DenseVector.java:133) > >> >> >> >> >> at > >> >> >> >> >> > >> >> >> >> >> > >> >> >> >> > >> >> >> > >> >> > >> > org.apache.mahout.math.decomposer.lanczos.LanczosSolver.solve(LanczosSolver.java:160) > >> >> >> >> >> at pca.PCASolver.solve(PCASolver.java:53) > >> >> >> >> >> at pca.PCA.main(PCA.java:20) > >> >> >> >> >> > >> >> >> >> >> So I should probably note that my data only has 2 columns, > >> the > >> >> >> real > >> >> >> >> data > >> >> >> >> >> will have quite a bit more. > >> >> >> >> >> > >> >> >> >> >> The failing happens with 10 and more for rank, with the > >> last, > >> >> and > >> >> >> >> >> therefore most significant eigenvector being <NaN,NaN>. > >> >> >> >> >> > >> >> >> >> >> -Trevor > >> >> >> >> >> > The 0 eigenvalue output is not valid, and yes, the output > >> >> will > >> >> >> list > >> >> >> >> >> the > >> >> >> >> >> > results > >> >> >> >> >> > in *increasing* order, even though it is finding the > >> largest > >> >> >> >> >> > eigenvalues/vectors > >> >> >> >> >> > first. > >> >> >> >> >> > > >> >> >> >> >> > Remember that convergence is gradual, so if you only ask > >> for > >> >> 3 > >> >> >> >> >> > eigevectors/values, you won't be very accurate. If you > >> ask > >> >> for > >> >> >> 10 > >> >> >> >> or > >> >> >> >> >> > more, > >> >> >> >> >> > the > >> >> >> >> >> > largest few will now be quite good. If you ask for 50, > >> now > >> >> the > >> >> >> top > >> >> >> >> >> 10-20 > >> >> >> >> >> > will > >> >> >> >> >> > be *extremely* accurate, and maybe the top 30 will still > >> be > >> >> >> quite > >> >> >> >> >> good. > >> >> >> >> >> > > >> >> >> >> >> > Try out a non-distributed form of what is in the > >> >> >> >> EigenverificationJob > >> >> >> >> >> to > >> >> >> >> >> > re-order the output and collect how accurate your results > >> are > >> >> >> (it > >> >> >> >> >> computes > >> >> >> >> >> > errors for you as well). > >> >> >> >> >> > > >> >> >> >> >> > -jake > >> >> >> >> >> > > >> >> >> >> >> > 2011/6/23 <[email protected]> > >> >> >> >> >> > > >> >> >> >> >> >> So, I know that MAHOUT-369 fixed a bug with the > >> distributed > >> >> >> >> version > >> >> >> >> >> of > >> >> >> >> >> >> the > >> >> >> >> >> >> LanczosSolver but I am experiencing a similar problem > >> with > >> >> the > >> >> >> >> >> >> non-distributed version. > >> >> >> >> >> >> > >> >> >> >> >> >> I send a dataset of gaussian distributed numbers (testing > >> >> PCA > >> >> >> >> stuff) > >> >> >> >> >> and > >> >> >> >> >> >> my eigenvalues are seemingly reversed. Below I have the > >> >> output > >> >> >> >> given > >> >> >> >> >> in > >> >> >> >> >> >> the logs from LanczosSolver. > >> >> >> >> >> >> > >> >> >> >> >> >> Output: > >> >> >> >> >> >> INFO: Eigenvector 0 found with eigenvalue 0.0 > >> >> >> >> >> >> INFO: Eigenvector 1 found with eigenvalue > >> 347.8703086831804 > >> >> >> >> >> >> INFO: LanczosSolver finished. > >> >> >> >> >> >> > >> >> >> >> >> >> So it returns a vector with eigenvalue 0 before one with > >> an > >> >> >> >> >> eigenvalue > >> >> >> >> >> >> of > >> >> >> >> >> >> 347?. Whats more interesting is that when I increase the > >> >> rank, > >> >> >> I > >> >> >> >> get > >> >> >> >> >> a > >> >> >> >> >> >> new > >> >> >> >> >> >> eigenvector with a value between 0 and 347: > >> >> >> >> >> >> > >> >> >> >> >> >> INFO: Eigenvector 0 found with eigenvalue 0.0 > >> >> >> >> >> >> INFO: Eigenvector 1 found with eigenvalue > >> 44.794928654801566 > >> >> >> >> >> >> INFO: Eigenvector 2 found with eigenvalue > >> 347.8286920203704 > >> >> >> >> >> >> > >> >> >> >> >> >> Shouldn't the eigenvalues be in descending order? Also is > >> >> the > >> >> >> 0.0 > >> >> >> >> >> >> eigenvalue even valid? > >> >> >> >> >> >> > >> >> >> >> >> >> Thanks, > >> >> >> >> >> >> Trevor > >> >> >> >> >> >> > >> >> >> >> >> >> > >> >> >> >> >> > > >> >> >> >> >> > >> >> >> >> >> > >> >> >> >> >> > >> >> >> >> > > >> >> >> >> > >> >> >> >> > >> >> >> >> > >> >> >> > > >> >> >> > >> >> >> > >> >> >> > >> >> > > >> >> > >> >> > >> >> > >> > > >> > >> > >> > > > > >
