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