TV norm prox

[fabian-sp]

Have you included the prox of the total variation norm, i.e. sum_i |x_i - x_{i-1}|?

I have implemented Condats algorithm in order to compute this (https://lcondat.github.io/publis/Condat-fast_TV-SPL-2013.pdf).
I could share my code.

Cheers!

[lostella]

I believe it’s not there in the package. Does the prox have closed form? Is it just for 1D signals or also 2D?

A PR is of course welcome :-) but let me know in case you need guidance there

[fabian-sp]

No, the prox is computed iteratively (see the referenced article). I think there is also a 2D version which would be useful for imaging problems I guess, but have not looked at that yet.
I would need at least some guidance on where to add the function :)

[lostella]

Of course! You can just add a source file in https://github.com/JuliaFirstOrder/ProximalOperators.jl/tree/master/src/functions say named tvnorm.jl and have the function type defined there. You can take a look at other source files there on how to organize it.

Only the call and prox! methods are strictly required (for other types, a prox_naive function is defined for testing purposes, but it maybe not needed in this case since there’s really no naive way of computing prox).

[fabian-sp]

Not sure whether you are interested in this, but one could also add some sum of norms, e.g. 1norm+2norm or 1norm+TV. Those are used in Group Lasso or Fused Lasso for example and the proximal operators can be shown to be just the concatenation of the individual proxes (in the right order).

[lostella]

@fabian-sp do you have references for those?

[fabian-sp]

1norm + 2norm: Proposition 2.1 in https://arxiv.org/pdf/1712.05910.pdf
1norm + TV: Proposition 4 in https://arxiv.org/pdf/1706.08732.pdf