Describe the bug

Forgive me if I've misunderstood what the wasserstein_circle function is supposed to do. But I would have thought that if we have

d1 = wasserstein_circle(arr1, arr2)

And then we add the same amount to both arrays (i.e. rotating both samples the same angle around the circle), then we should get the same answer.

d2 = wasserstein_circle(arr1 + delta, arr2 + delta)
assert d1 == d2

But the first example I tried fails.

To Reproduce

import numpy as np
import ot

sample1 = np.array([0.1, 0.11, 0.4, 0.6])
sample2 = np.array([0.21, 0.15, 0.7, 0.95])

d1 = ot.wasserstein_circle(sample1, sample2)

delta = 0.02

d2 = ot.wasserstein_circle(sample1 + delta, sample2 + delta)

assert d1 == d2 # fails

Expected behavior

wasserstein_circle should be rotationally symmetric, i.e. it should obey the property

d1 = wasserstein_circle(arr1, arr2)
d2 = wasserstein_circle((arr1 + delta) % 1, (arr2 + delta) % 1)
assert d1 == d2

For all real delta (up to floating point inaccuracies), because this amounts to just turning your head to the side.

Or am I just misunderstanding how the input is supposed to be represented here?

Environment (please complete the following information):

• OS (e.g. MacOS, Windows, Linux): NixOS 24.11
• Python version: 3.12.7
• How was POT installed (source, pip, conda): Nix

Output of the following code snippet:

import platform; print(platform.platform())
import sys; print("Python", sys.version)
import numpy; print("NumPy", numpy.__version__)
import scipy; print("SciPy", scipy.__version__)
import ot; print("POT", ot.__version__)

Linux-6.6.67-x86_64-with-glibc2.40
Python 3.12.7 (main, Oct  1 2024, 02:05:46) [GCC 13.3.0]
NumPy 1.26.4
SciPy 1.14.1
POT 0.9.4

Additional context