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 >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> > >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> > >> >> >> >> >> >> >> >> >> >> >> >> >> >> >> > >> >> >> >> >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > >> >> >> >
