It Gramian is at least positive semidefinite and will be definite if the
matrix is non singular, yes. That's usually but not always true.
The lambda*I matrix is positive definite, well, when lambda is positive.
Adding that makes it definite.
At least, lambda=0 could be rejected as invalid.
But this goes back to using the Cholesky decomposition. Why not use QR? It
doesn't require definite. It should be a little more accurate. On these
smallish dense matrices I don't think it is much slower. I have not
benchmarked that but I opted for QR in a different implementation and it
has worked fine.
Now I have to go hunt for how the QR decomposition is exposed in BLAS...
Looks like its GEQRF which JBLAS helpfully exposes. Debasish you could try
it for fun at least.
On Oct 15, 2014 8:06 PM, "Debasish Das" <debasish.da...@gmail.com> wrote:
> But do you expect the mllib code to fail if I run with 0.0 regularization ?
>
> I think ||r - wi'hj||^{2} is positive definite...It can become positive
> semi definite only if there are dependent rows in the matrix...
>
> @sean is that right ? We had this discussion before as well...
>
>
> On Wed, Oct 15, 2014 at 5:01 PM, Liquan Pei <liquan...@gmail.com> wrote:
>
> > Hi Debaish,
> >
> > I think ||r - wi'hj||^{2} is semi-positive definite.
> >
> > Thanks,
> > Liquan
> >
> > On Wed, Oct 15, 2014 at 4:57 PM, Debasish Das <debasish.da...@gmail.com>
> > wrote:
> >
> >> Hi,
> >>
> >> If I take the Movielens data and run the default ALS with regularization
> >> as
> >> 0.0, I am hitting exception from LAPACK that the gram matrix is not
> >> positive definite. This is on the master branch.
> >>
> >> This is how I run it :
> >>
> >> ./bin/spark-submit --total-executor-cores 1 --master spark://
> >> tusca09lmlvt00c.uswin.ad.vzwcorp.com:7077 --jars
> >>
> >>
> /Users/v606014/.m2/repository/com/github/scopt/scopt_2.10/3.2.0/scopt_2.10-3.2.0.jar
> >> --class org.apache.spark.examples.mllib.MovieLensALS
> >> ./examples/target/spark-examples_2.10-1.1.0-SNAPSHOT.jar --rank 20
> >> --numIterations 20 --lambda 0.0 --kryo
> >> hdfs://localhost:8020/sandbox/movielens/
> >>
> >> Error from LAPACK:
> >>
> >> WARN TaskSetManager: Lost task 0.0 in stage 11.0 (TID 22,
> >> tusca09lmlvt00c.uswin.ad.vzwcorp.com):
> >> org.jblas.exceptions.LapackArgumentException: LAPACK DPOSV: Leading
> minor
> >> of order i of A is not positive definite.
> >>
> >> From the maths it's not expected right ?
> >>
> >> ||r - wi'hj||^{2} has to be positive definite...
> >>
> >> I think the tests are not running any 0.0 regularization tests otherwise
> >> we
> >> should have caught it as well...
> >>
> >> For the sparse coding NMF variant that I am running, I have to turn off
> L2
> >> regularization when I run a L1 on products to extract sparse topics...
> >>
> >> Thanks.
> >>
> >> Deb
> >>
> >
> >
> >
> > --
> > Liquan Pei
> > Department of Physics
> > University of Massachusetts Amherst
> >
>
