Last updated: 2025-06-24
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Knit directory:
symmetric_covariance_decomposition/
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This site is for further exploring symmetric covariance decomposition methods.
Main takeways thus far (updated June 24):
symEBcovMF with backfit with point-exponential prior also does relatively well in the tree setting. However, the representation is not fully sparse (We also see this with GBCD and symEBcovMF with the point-Laplace plus splitting initialization). I suspect this is due to the model misspecification with regards to the noise. The corresponding analysis is here.
symEBcovMF backfit initialized with the point-Laplace plus splitting strategy does relatively well in the tree setting. One observation is the representation is not fully sparse; some of the factors have small loadings in other populations alongside the main component. The corresponding analysis is here.
When applying symEBcovMF with point-Laplace prior in the tree
setting, the greedy method does not find the sparse representation.
However, if you backfit for long enough, e.g. 20,000 iterations,
symEBcovMF eventually will find the sparse representation. This behavior
is also seen in flashier and EBCD. The corresponding analysis is here
I think the greedy methods do not find the sparse representation
because \(F\) is not exactly orthogonal
(I generate the data matrix \(Y\) as
\(Y = LF' + E\) where \(F_{ij} \overset{i.i.d.}{\sim} N(0,2^2)\)).
It seems like the method is picking up on correlations between the
factors. My investigation of this can be found here
Flashier’s backfit is a lot faster than symEBcovMF’s backfit because of its extrapolation technique
For symEBcovMF, refitting the lambda values after a new factor is
added helps improve the fit. This is seen in the balanced,
nonoverlapping analysis here. It can also help the
method find a more complete representation. This is especially apparent
in the tree setting analysis here.
For the sparse overlapping setting, the method performs better when the Kmax parameter is set to a larger number. This is seen in the analysis here.
symEBcovMF analyses:
Analyses related to initialization:
Tree residual matrix example: