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From tr...@cs.drexel.edu
Subject Re: LanczosSVD and Eigenvalues
Date Thu, 23 Jun 2011 18:39:08 GMT
```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 <tra26@cs.drexel.edu>
>
>> 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 <tra26@cs.drexel.edu>
>> >
>> >> 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 <tra26@cs.drexel.edu>
>> >> >
>> >> >> 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 <tra26@cs.drexel.edu>
>> >> >> >
>> >> >> >> 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
>> 3
>> >> >> >> > eigevectors/values, you won't be very accurate.
>> 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 <tra26@cs.drexel.edu>
>> >> >> >> >
>> >> >> >> >> 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
>> >> >> >> >>
>> >> >> >> >>
>> >> >> >> >
>> >> >> >>
>> >> >> >>
>> >> >> >>
>> >> >> >
>> >> >>
>> >> >>
>> >> >>
>> >> >
>> >>
>> >>
>> >>
>> >
>>
>>
>>
>

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