How to use own low rank types for A and B

[mipals]

I was wondering which operations do you need to have implemented for A and B in order to use your implementation?

Sorry if this should not be posted as an issue. Im pretty new to GitHub.

[dlfivefifty]

At the moment SemiseparableMatrix is not really implemented, but it supports any matrix types for L and U.

AlmostBandedMatrix has it hard-coded as SemiseparableMatrices.LowRankMatrix, which is a type alias for ApplyMatrix(*,U,V). This could be redesigned. The code itself just uses U,V = arguments(fillpart(A)) so this could be changed to an interface. Though perhaps we would want to support 3-argument low-rank formats (U*Σ*V where Σ is diagonal)?

[dlfivefifty]

Sorry if this should not be posted as an issue. Im pretty new to GitHub.

Issues are the best place to have these types of conversations, thanks for posting!

[mipals]

Thanks a lot for the clarification. Do you have any future plans to further develop the capabilities of SemiseparableMatrix?

I dont think its particularly clear from the description that AlmostBandedMatrix is hard-coded with B being a LowRankMatrix, but i guess it makes sense if the code uses U and V directly. Supporting 3-argument low-rank formats could be useful, but I guess it might suffice to just "include" Σ in U and V?

[dlfivefifty]

Thanks a lot for the clarification. Do you have any future plans to further develop the capabilities of SemiseparableMatrix?

No concrete plans, so I would love to have help! The feature I want most is for representing inverses of banded matrices as Banded + Semiseparable, I'm pretty sure it can be done in O(n).

I dont think its particularly clear from the description that AlmostBandedMatrix is hard-coded with B being a LowRankMatrix

This was just quickly thrown together (hence v0.0.1).

Supporting 3-argument low-rank formats could be useful, but I guess it might suffice to just "include" Σ in U and V?

Right, we can always return U, Σ*V or in lazy form U, ApplyMatrix(*, Σ, V). This would be really easy for anything that supports the ApplyArray interface (in fact sub-views of ApplyArray do and already work):

_two_arguments(A,B) = (A,B)
_two_arguments(A,B, C...) = (A, ApplyMarix(*, B, C...))
two_arguments(A) = _two_arguments(arguments(A)...)

[mipals]

No concrete plans, so I would love to have help! The feature I want most is for representing inverses of banded matrices as Banded + Semiseparable, I'm pretty sure it can be done in O(n).

I have only worked with symmetric extended generator representable semiseparable matrices (symmetric EGRSS matrices) which have the form tril(UV') + triu(VU',1). For these Cholesky factorizations (and its inverse) have a similar form L = tril(UW') (L^{-1} = tril(YZ') ) and can be computed in O(n). Similar results happens when you add a diagonal matrix tril(UV') + triu(VU',1) + diag(d), so I guess equivalent speeds should be possible for ```Semiseparable + Banded``.

Right, we can always return U, Σ*V or in lazy form U, ApplyMatrix(*, Σ, V). This would be really easy for anything that supports the ApplyArray interface (in fact sub-views of ApplyArray do and already work):

_two_arguments(A,B) = (A,B)
_two_arguments(A,B, C...) = (A, ApplyMarix(*, B, C...))
two_arguments(A) = _two_arguments(arguments(A)...)

That was exactly what i was thinking. I dont know any particulars of the ApplyArray interface, but from you say it does look like a viable route to take.

[dlfivefifty]

Note https://github.com/JuliaMatrices/LowRankApprox.jl which has a lot of low rank approximations. The only reason I'm not using that is I needed support for views of a low rank matrix to also conform to the "low rank interface", which is true for AppliedArray but not for LowRankApprox.LowRankMatrix.

I should unify this a bit better.