Hi Christian (and hi list),
On Wed, Mar 16, 2011 at 11:18:18PM +0100, Christian F. Beckmann wrote:
> No, the current release version does not run multiple runs. While your
> list of references indeed discuss benefits of combining multiple runs
> the field in general has forgotten relevant papers by Rao which show
> that the ICA solution is unique (see our 2004 paper for a discussion)
To add a minor precision, the uniqueness of a solution does not guaranty
that a given algorithm will find it. In particular, when the algorithm is
solving a non-convex optimization problem, the algorithm, unless it is
doing a global search, will likely get stuck in local optimum which do
not correspond to the global optimum and thus not the unique solution of
the problem. In this situation, the initial point of optimization will
condition the local minimum found.
This is one of the reasons why a package like ICASSO exists. The other
one is that ICASSO enables (I believe) bootstrapping of the data, which
is also useful for algorithms that guaranties convergence to the unique
solution, if they don't give bounds on the variability of this solution
as a function of the data (eg Lipshich-continuous properties).
Both problems occur for ICA: it is not a convex problem in general, and
its solution can show large (arbitrary large, if I am not mistaken)
dependence on the data. However, the good news is that if the Kurtosis is
used as a contrast function, as I believe it is the case by default in
Melodic, the problem is convex, and FasICA is able to retrieve the global
optimal (see for instance [1], the paragraph between formulas 20 and 21).
Thus running Melodic multiple times with different initialization
shouldn't change anything to the results, as Christian mentioned.
Sorry if I am being pedantic, I thought that shedding some light on
the motivations behind ICASSO could enrich the discussion.
Cheers,
Gael
[1] Hyvarinen 1999, 'Fast and robust fixed-point algorithms for
independence component analysis'
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