Jacobian rank determination in LevenbergMarquardtOptimizer is not numerically
robust
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Key: MATH-352
URL: https://issues.apache.org/jira/browse/MATH-352
Project: Commons Math
Issue Type: Bug
Affects Versions: 2.0
Environment: commons-math 2.0
Reporter: Gene Gorokhovsky
Priority: Critical
Fix For: 2.1
LevenbergMarquardtOptimizer is designed to handle singular jacobians, i.e.
situations when some of the fitted parameters depend on each other. The check
for that condition is in LevenbergMarquardtOptimizer.qrDecomposition uses
precise comparison to 0.
if (ak2 == 0 ) {
rank = k;
return;
}
A correct check would be comparison with a small epsilon. Hard coded 2.2204e-16
is used elsewhere in the same file for similar purpose.
final double QR_RANK_EPS = Math.ulp(1d); //2.220446049250313E-16
....
if (ak2 < QR_RANK_EPS) {
rank = k;
return;
}
Current exact equality check is not tolerant of the real world poorly
conditioned situations. For example I am trying to fit a cylinder into sample
3d points. Although theoretically cylinder has only 5 independent variables,
derivatives for optimizing function (signed distance) for such minimal
parametrization are complicated and it it much easier to work with a 7
variable parametrization (3 for axis direction, 3 for axis origin and 1 for
radius). This naturally results in rank-deficient jacobian, but because of the
numeric errors the actual ak2 values for the dependent rows ( I am seeing
values of 1e-18 and less), rank handling code does not kick in.
Keeping these tiny values around then leads to huge corrections for the
corresponding very slowly changing parameters, and consequently to numeric
errors and instabilities. I have noticed the problem because tiny shift in the
initial guess (on the order of 1e-12 in the axis component and origins)
resulted in significantly different finally converged answers (origins and
radii differing by as much as 0.02) which I tracked to loss of precision due to
numeric error with root cause described above.
Providing a cutoff as suggested fixes the issue. After the fix, small
perturbations in the initial guess had practically no effect to the converged
result - as expected from a robust algorithm.
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