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From Gilles Sadowski <gil...@harfang.homelinux.org>
Subject Re: [Math] "LUDecomposition" in "AbstractLeastSquaresOptimizer"
Date Thu, 08 Sep 2011 14:09:18 GMT
On Thu, Sep 08, 2011 at 12:15:20AM -0700, Phil Steitz wrote:
> On 9/7/11 6:34 AM, Gilles Sadowski wrote:
> > Hi.
> >
> >>>>> In class "AbstractLeastSquaresOptimizer" (in "o.a.c.m.optimization.general"),
> >>>>> the method "getCovariances()" uses "LUDecompositionImpl" to compute
the
> >>>>> inverse of a matrix.
> >>>>> In my application, this leads to a "SingularMatrixException". If
I change
> >>>>> "LUDecompositionImpl" to "QRDecompositionImpl", no exception is
raised.
> >>>>> Also, keeping "LUDecompositionImpl" but passing a much lower singularity
> >>>>> threshold, does not raise the exception either.
> >>>>>
> >>>>> Thus, I wonder whether there was a reason for using "LU", and if
not,
> >>>>> whether I could change the decomposition solver to "QR" (as this
is a
> >>>>> cleaner solution than guessing a good value for the threshold).
> >>>> There are no reason for LU decomposition, and QR decomposition is
> >>>> known to be more stable. So I would also consider switching to this
> >>>> algorithm is a cleaner solution.
> >>> Fine. I'll open a JIRA issue.
> >>>
> >>> A unit test "testNonInvertible" in "LevenbergMarquardtOptimizerTest" fails
> >>> with the change to "QRDecomposition" because no "SingularMatrixException"
> >>> is raised anymore.
> >>> What was the purpose of that test?
> >> The purpose was to check that impossible problems were detected properly.
> > My question should have been clearer: Was the previous behaviour correct
> > (i.e. an exception *must* be raised somehow)?
> 
> Yes.  The getCovariances method is trying to invert a singular
> matrix.  SingularMatrixException is appropriate there.

Still, I don't understand: The "problem" is passed to the optimizer which
returns an answer. Is the answer correct? If not, what if the user does not
call "getCovariances()"?
The failure should be detected earlier (possibly by the optimization
algorithm). And if it is not possible during the search, shouldn't there be
a utility method explicitly aimed at checking the quality of the solution?

> The problem
> is that the QR decomp method has no threshold defined for
> singularity, doing an exact check against 0 on the elements of
> rDiag.  Many (most?) singular matrices or
> near-singular-enough-to-lead-to-meaningless-results matrices will
> not lead to exact 0's there due to rounding in the computations. 
> That's why the LU impl defines a threshold.   For the (exactly)
> singular matrix in the test case above, LU does not get exact 0's
> either on pivots, but the default singularity threshold (correctly)
> detects the singularity.  IMO, QR should do the same thing;
> otherwise we will happily return meaningless results, as in this
> case.  In this case, there is no question, since the matrix is
> exactly singular.  In cases where matrices are near-singular, we are
> doing users a favor by throwing, since inversion results are going
> to be unstable.
> 
> A possibly more robust option here is to use Cholesky decomposition,
> which is known to be stable for symmetric positive definite
> matrices, which the covariance matrix being inverted here should
> be.  The exceptions thrown will be different; but they will give
> more specific information about what is wrong with the covariance
> matrix.

I've tried it with my problem, and it also throws an exception.
However, I would like to obtain the covariance matrix anyway, because I've
no other clue as to what might be wrong.
So I think that, at least, users should be able to set the positive
definiteness threshold in order to avoid raising an exception.

> 
> Luc - are there other reasons that QR would be better for cov
> matrices?   I would have to play with a bunch of examples, but I
> suspect with the defaults, Cholesky may do the best job detecting
> singular problems. 

What do you mean by "with the defaults"?


Regards,
Gilles

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