2011/6/23 <[email protected]> > That times method won't get me past what you described in: > http://article.gmane.org/gmane.comp.apache.mahout.user/7384/match=pca > > with 2 dense matrices being multiplied will it? And it is conceivable that > we will have billions of rows to do this computation on for my dataset. >
your matrix is already dense, is it not? mean subtraction won't make it any *more* dense. > On another note, is that not a confusing function name as it does not do > just multiplication but transpose on the matrix as well? > That function name is indeed confusing, and it does effectively does a transpose on the matrix doing the method call. A.times(B) is really A.transpose().multipliedBy(B), but it does it all in one step. -jake > -Trevor > > Well I'm going to pretend for a second that you're talking about > > *billions* > > of rows, because tens of millions of rows still fit not just on one box, > > but > > on one box *in memory*... but yes, the mapreduce job you want already > > exists in Mahout: DistributedRowMatrix#times(DistributedRowMatrix) > > does exactly this. > > > > Although it doesn't do the mean-subtraction. > > > > -jake > > > > 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 > >> >> >> >> >> >> >> > >> >> >> >> >> >> >> > >> >> >> >> >> >> > > >> >> >> >> >> >> > >> >> >> >> >> >> > >> >> >> >> >> >> > >> >> >> >> >> > > >> >> >> >> >> > >> >> >> >> >> > >> >> >> >> >> > >> >> >> >> > > >> >> >> >> > >> >> >> >> > >> >> >> >> > >> >> >> > > >> >> >> > >> >> >> > >> >> >> > >> >> > > >> >> > >> >> > >> >> > >> > > >> > >> > >> > > > > >
