Diffusion Earth Mover's Distance embeds the Wasserstein distance between two distributions on a graph into L^1 in log-linear time.

MultiscaleEMD is available in pypi. Install by running the following:

pip install MultiscaleEMD

MultiscaleEMD is written following the sklearn estimator framework. We provide two functions that operate quite differently. First the Chebyshev approxiamtion of the operator in DiffusionCheb, which we recommend when the number of distributions is small compared to the number of points. Second, the Interpolative Decomposition method that computes dyadic powers of $P^{2^k}$ directly in DiffusionTree. These two classes are used in the same way, first supplying parameters, fitting to a graph and array of distributions:

import numpy as np
from DiffusionEMD import DiffusionCheb

# Setup an adjacency matrix and a set of distributions to embed
adj = np.ones((10, 10))
distributions = np.random.randn(10, 5)
dc = DiffusionCheb()

# Embeddings where the L1 distance approximates the Earth Mover's Distance
embeddings = dc.fit_transform(adj, distributions)
# Shape: (5, 60)

Additionally, there are other multiscale structures that can be used to

Requirements can be found in requirements.txt

Examples are in the notebooks directory.

Take a look at the examples provided there to get a sense of how the parameters behave on simple examples that are easy to visualize.

This code implements the algorithms described in this paper:

ArXiv Link: http://arxiv.org/abs/2102.12833:

@InProceedings{pmlr-v139-tong21a,
  title =       {Diffusion Earth Mover’s Distance and Distribution Embeddings},
  author =      {Tong, Alexander Y and Huguet, Guillaume and Natik, Amine and Macdonald, Kincaid and Kuchroo, Manik and Coifman, Ronald and Wolf, Guy and Krishnaswamy, Smita},
  booktitle =   {Proceedings of the 38th International Conference on Machine Learning},
  pages =       {10336--10346},
  year =        {2021},
  editor =      {Meila, Marina and Zhang, Tong},
  volume =      {139},
  series =      {Proceedings of Machine Learning Research},
  month =       {18--24 Jul},
  publisher =   {PMLR},
  pdf =         {http://proceedings.mlr.press/v139/tong21a/tong21a.pdf},
  url =         {http://proceedings.mlr.press/v139/tong21a.html},
}

And this paper:

ArXiv Link: https://arxiv.org/abs/2107.12334

As well as other algorithms under development.