More general columnwise multiplication action

[olivierverdier]

This makes ColumnwiseMultiplicationAction more general, as it works now with any subtype of AbstractMatrixGroup, a new abstract type above GeneralLinear and GeneralUnitary.

Feel free to change the new name if you don't like it!

Maybe other groups could be subgroups of this new abstract type?

I also think more actions could be defined with respect to that abstract type instead of GeneralUnitary.

What do you think?

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 96.38%. Comparing base (41e170d) to head (66fb5c1).
Report is 1 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff           @@
##           master     #756   +/-   ##
=======================================
  Coverage   96.38%   96.38%           
=======================================
  Files         123      123           
  Lines       11438    11438           
=======================================
  Hits        11025    11025           
  Misses        413      413           

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[mateuszbaran]

Thanks, I think this is a good idea.

One thing I'm unsure about at the moment is that it also makes sense for SE(n) to be able to act through ColumnwiseMultiplicationAction but not in a straightforward way and we can't make special Euclidean a subtype of AbstractMatrixGroup. So maybe ColumnwiseMultiplicationAction should be even more general and allow for arbitrary On, and some groups would have to specialize apply for this action.

[olivierverdier]

Yes, that's a good point, I agree with that.

Maybe something else is required for that: a diagonal action, that is, when you have an action of $G$ on a space $V$, you should be able to construct the diagonal action of $G$ on $V^n$ for some integer $n$.

Then the columnwise action on $d\times n$ matrices is essentially a diagonal action.

That would solve the issue in the most general way, I believe?

So a rough sketch of how it would look like is:

struct ColumnwiseMultiplicationAction{
    TA<:AbstractGroupAction{TAD}
} <: AbstractGroupAction{TAD}
    action::TA
end

function apply(A::ColumnwiseMultiplicationAction, a, p)
    return apply(DiagonalAction(A.action, size(p,2)), a, p) # but DiagonalAction doesn't exist yet (and not sure if `p` or `eachcol(p)`)
end

[kellertuer]

Just a first super small remark. In LieGroups.jl there is only one (concrete) type LieGroup{Field, Manifold, GroupOperation} would a MatrixLieGroup there make sense as well? Then we need AbstractLieGroup somewhen there (maybe anyways) – or is it more useful to do an AbstractMatrixMultiplicationGroupOperation since we have a MultiplicationGroupOperation already anyways?

edit: there is also no need to hurry, in the current speed and with just me writing on Lie groups. I would like to finish most groups and their operations first, so until I come to concrete actions in that speed, is probably still quite a few weeks to go.

[mateuszbaran]

Maybe something else is required for that: a diagonal action, that is, when you have an action of

Good idea, I will think about it.

Just a first super small remark. In LieGroups.jl there is only one (concrete) type LieGroup{Field, Manifold, GroupOperation} would a MatrixLieGroup there make sense as well? Then we need AbstractLieGroup somewhen there (maybe anyways) – or is it more useful to do an AbstractMatrixMultiplicationGroupOperation since we have a MultiplicationGroupOperation already anyways?

At the moment it's not entirely clear what that abstract type would exactly mean. I think we can implement the key idea here without that type anyway, so maybe there is no need to think about implementing it in LieGroups.jl.

[olivierverdier]

Yes, I suppose it could make sense for LieGroups.jl as well?

But I'm not 100% sure that this is the correct way to handle the issue in this case, since, as @mateuszbaran observed, multicolumn action should also be available for direct and semidirect product, and the current solution doesn't address these cases. I honestly don't know what the best solution would be.

[kellertuer]

Sure, taking this as a bit of a playground and experiment sounds good. I feel it might be nicer to introduce an AbstractMultiplicationOp and a MatrixOp or so on the operations side and subtype there. But I have not yet fully thought through that and all ist implications either.

[mateuszbaran]

I've switched AbstractMatrixGroup to a union type MatrixGroup for now. This should now be a simple extension that won't need too much consideration and we can think about something more general for LieGroups.jl later. Is that OK?

[olivierverdier]

Yes, I think that will do, thank you.