Feat/logeuclid mean and logeuclid distance to convex hull

[gcattan]

This PR implements the logeuclid mean with constraint programming model, and fixes logeuclid distance to convex hull.

[toncho11]

Do you have first results?

[gcattan]

Yes, and it is not working. I guess it will not be that easy to implement ^^

[gcattan]

@qbarthelemy what do you think of this implementation. Am I missing something?

[toncho11]

@qbarthelemy So it is as the previous implementation but now logm is applied on the COV matrices, then the definition with constrain programming and in the end expm. So it like in pyRiemann:
https://github.com/pyRiemann/pyRiemann/blob/9214eb3e1d00c51210c2886cc7e5fc8a54dbc7ef/pyriemann/utils/mean.py#L332
Is it OK?

[gcattan]

Thanks for the reviewer @qbarthelemy . I will address the comments as soon as possible.
I did some tests today with the new log-euclidian distance.

I see a difference in the matrices, whether I do:
logm(covmats) or np.array([logm(covmat) for covmat in covmats])

The first option returns matrices with mostly zeros.

Then, when I take the exponential of the mean returned by the optimizer, it is always a diagonal matrix.

I suspect there is some issue with the optimizer. I see already that Adam is providing better results than Cobyla, when debugging with a classical optimizer.

[gcattan]

Hm. The equation should be minimal when the sum of the covariance matrices (Xi) leans towards Y.
I will add the constraint, but this could be an issue for QAOA, as it should only accept quadratic and unconstraint binary problems. Let's see if he can manage this.

I cannot remember why this 1 - x to be honest. It looks like the prediction of the MDM was inverted, but I cannot remember why. I moved this pre-processing outside the function for the moment.

[qbarthelemy]

Hm. The equation should be minimal when the sum of the covariance matrices (Xi) leans towards Y.

Yes, you are right. But null weights give the same minimal objective.
Before merging, can you add check_weights on master version to check if weights were always null, or not?

Furthermore, the objective can become even smaller if weights are negative.
Is it possible to add in the optimizer a constraint on positivity (like in Eq(22) of Zhao2019)?

[qbarthelemy]

In fact, tests fail raising the new checking Weights must be strictly positive

https://github.com/pyRiemann/pyRiemann-qiskit/actions/runs/8061607519/job/22019679900#step:7:401

[gcattan]

We can still set the lower bound to some minimum, such as 10^-4.

However, may be in this case, the pb is with the matrices that are generated for the test.
We can use mne data as it is done for distance.

Actually, this test here is redundant:

It should be located (and is already implemented in test_utils_distance.py)