LMNN: fix mistake and improve performances

[toto6]

Fix mistake in LMNN
Issue in function _find_impostors:

• the squared euclidean distance is used to compute the margins in variable "margin_radii"
• the euclidean distance is used (through the function sklearn.metrics.pairwise.pairwise_distances) to compute distances between samples of different labels in variable "dist"
• the issue is that the impostors are found by testing "dist < margin_radii" which is wrong because "dist" represent distances, and "margin_radii" represent squared distances.

I propose to solve this problem by computing always the squared distances.

Faster LMNN
The use of the function sklearn.metrics.pairwise_distances gives horrible performances.
Replace it by a faster function.

Below is a little script to show the computation time of LMNN

import time
from sklearn.datasets import load_breast_cancer
from metric_learn import lmnn
dataset = load_breast_cancer()
s = time.time()
l1 = lmnn.LMNN(k=3)
l1.fit(dataset["data"], dataset["target"])
e = time.time()
print(e - s)

Before this commit, this script executes in 32 seconds
After this commit, this script executes in 1 second

[perimosocordiae]

Thanks for the contribution! I agree with your analysis of the bug, though I was surprised to hear that sklearn's pairwise_distances was so slow.

So I looked a little closer and found that the issue is actually metric='seuclidean'. Regular L2 distance uses essentially the same algorithm as your pairwiseEuclidean:
https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/metrics/pairwise.py#L246

But if you use the seuclidean metric, it ends up calling Scipy's cdist function instead, which then has to make a copy of both input arrays to make them C-contiguous:
https://github.com/scipy/scipy/blob/v1.0.0/scipy/spatial/distance.py#L2361

So I think the easiest solution is, instead of rolling our own distance function, importing euclidean_distances from sklearn and calling it with squared=True anywhere that we want squared L2 distances. I did some quick testing and found that this also prevents the very slow running time.

[perimosocordiae]

You can leave out the second argument, similar to how pairwise_distances works.