issue with using other metric choices with emd2_1d

[mrunalimanj]

Describe the bug

emd2_1d errors when not using the sped-up distribution metrics, e.g. cosine, yule,

To Reproduce

Steps to reproduce the behavior:
Simple test case adapted from the 1d example code:

import numpy as np
import matplotlib.pylab as pl
import ot
import ot.plot
from ot.datasets import make_1D_gauss as gauss

##############################################################################
# Generate data
# -------------


#%% parameters

n = 100  # nb bins

# bin positions
x = np.arange(n, dtype=np.float64)

# Gaussian distributions
a = gauss(n, m=20, s=5)  # m= mean, s= std
b = gauss(n, m=60, s=10)

# use fast 1D solver
G0 = ot.emd_1d(x, x, a, b, metric="cosine")

 54 G0 = ot.emd_1d(x, x, a, b, metric="cosine")
     55 
     56 # Equivalent to

~/miniconda3/envs/ms-gen/lib/python3.8/site-packages/ot/lp/solver_1d.py in emd_1d(x_a, x_b, a, b, metric, p, dense, log, check_marginals)
    257     perm_b = nx.argsort(x_b_1d)
    258 
--> 259     G_sorted, indices, cost = emd_1d_sorted(
    260         nx.to_numpy(a[perm_a]).astype(np.float64),
    261         nx.to_numpy(b[perm_b]).astype(np.float64),

ot/lp/emd_wrap.pyx in ot.lp.emd_wrap.emd_1d_sorted()

AttributeError: 'float' object has no attribute 'reshape'

Expected behavior

Should return a value, but instead errors (can't tell if math is yet correct)

Environment (please complete the following information):

• OS (e.g. MacOS, Windows, Linux): Linux
• Python version: 3.8.18
• How was POT installed (source, pip, conda): pip
• Build command you used (if compiling from source): pip install POT

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-5.15.0-117-generic-x86_64-with-glibc2.10
Python 3.8.18 | packaged by conda-forge | (default, Oct 10 2023, 15:44:36)
[GCC 12.3.0]
NumPy 1.24.4
SciPy 1.10.1
POT 0.9.4

Additional context

[rtavenar]

Hi @mrunalimanj ,

The emd_1d and emd2_1d functions rely on the result presented (for example) in [1, Remark 2.28]. As such, it is valid only for metrics of the form $d(x, y) = |x - y|^p$ (at least, to the best of my knowledge).

I hence suggest that we remove the opportunity to select metrics that are not of this form, and I have opened a PR for that.

[rflamary]

The PR was merged I'm closing the issue