Add NormL1plusL2

[fabian-sp]

I created the function for the sum of L1 and L2. I think together with SlicedSeparableSum this is exactly what you need for the Group Lasso penalty.
Regarding tests: could add that it has to match NormL1 and NormL2 for one of the two parameters being zero, but was not so sure how/where to do that.
Another question: I had some problems importing the prox!-function from NormL1/2, and solved it with

import ProximalOperators: prox!

Is this the intended way in Julia to import functions from other files within a module?

Cheers,
Fabian

[codecov]

Codecov Report

Merging #133 (d2689ea) into master (fcd3d98) will decrease coverage by 0.13%.
The diff coverage is 70.00%.

@@            Coverage Diff             @@
##           master     #133      +/-   ##
==========================================
- Coverage   87.42%   87.29%   -0.14%     
==========================================
  Files          78       79       +1     
  Lines        2490     2510      +20     
==========================================
+ Hits         2177     2191      +14     
- Misses        313      319       +6     

Impacted Files	Coverage Δ
src/ProximalOperators.jl	68.75% <ø> (ø)
src/functions/normL1plusL2.jl	70.00% <70.00%> (ø)

---

Continue to review full report at Codecov.

Legend - Click here to learn more
Δ = absolute <relative> (impact), ø = not affected, ? = missing data
Powered by Codecov. Last update fcd3d98...d2689ea. Read the comment docs.

[mfalt]

I wasn't completely happy with the implementation of prox! so I made a suggestion in a PR to your branch: https://github.com/fabian-sp/ProximalOperators.jl/pull/1

I didn't like that we created the L1 and L2 norms in the prox! call because it forces us to recheck that the parameters are positive every time. I rewrote the struct to contain the norm functions instead, and allowed lambda1 to be a vector. I think the code looks slightly better this way.
This PR also defer almost all computations to existing implementations.

[mfalt]

I made a couple of changes, for example allowing complex domain, added more tests, and fixed the printing for Array case. I hope you don't mind.
I am not too happy about my implementation of prox_naive do we have a better way to verify all cases?
Can someone verify that the same rule works for complex arguments and vector lambla1?

[fabian-sp]

Hi Mattias, I am not sure whether the proof would work for complex lambda1 as it involves the subdifferential, and I have never encountered that for complex-valued function.

[fabian-sp]

Hi all, is this still open/ under discussion?

Best, Fabian

[lostella]

@fabian-sp this looks good to me! Thanks for implementing this!

[lostella]

Hi Mattias, I am not sure whether the proof would work for complex lambda1 as it involves the subdifferential, and I have never encountered that for complex-valued function.

I think the general setting of the package is that of (extended) real valued functions, so all coefficients in front of norms and penalties should have no imaginary part