It can be considered an ordered pair (M, d) such that d: M M . For regularized Optimal Transport, the main reference on the subject is Wasserstein Distance) for these two grayscale (299x299) images/heatmaps: Right now, I am calculating the histogram/distribution of both images. Compute the first Wasserstein distance between two 1D distributions. Default: 'none' Thanks for contributing an answer to Cross Validated! Copyright 2016-2021, Rmi Flamary, Nicolas Courty. Can corresponding author withdraw a paper after it has accepted without permission/acceptance of first author. Find centralized, trusted content and collaborate around the technologies you use most. a kernel truncation (pruning) scheme to achieve log-linear complexity. L_2(p, q) = \int (p(x) - q(x))^2 \mathrm{d}x Doing it row-by-row as you've proposed is kind of weird: you're only allowing mass to match row-by-row, so if you e.g. If you see from the documentation, it says that it accept only 1D arrays, so I think that the output is wrong. In that respect, we can come up with the following points to define: The notion of object matching is not only helpful in establishing similarities between two datasets but also in other kinds of problems like clustering. Wasserstein Distance-Based Nonlinear Dimensionality Reduction for Depth Update: probably a better way than I describe below is to use the sliced Wasserstein distance, rather than the plain Wasserstein. Since your images each have $299 \cdot 299 = 89,401$ pixels, this would require making an $89,401 \times 89,401$ matrix, which will not be reasonable. Asking for help, clarification, or responding to other answers. Consider two points (x, y) and (x, y) on a metric measure space. MathJax reference. When AI meets IP: Can artists sue AI imitators? Metric measure space is like metric space but endowed with a notion of probability. If it really is higher-dimensional, multivariate transportation that you're after (not necessarily unbalanced OT), you shouldn't pursue your attempted code any further since you apparently are just trying to extend the 1D special case of Wasserstein when in fact you can't extend that 1D special case to a multivariate setting. be solved efficiently in a coarse-to-fine fashion, v(N,) array_like. Making statements based on opinion; back them up with references or personal experience. It could also be seen as an interpolation between Wasserstein and energy distances, more info in this paper. Right now I go through two libraries: scipy (https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.wasserstein_distance.html) and pyemd (https://pypi.org/project/pyemd/). If you downscaled by a factor of 10 to make your images $30 \times 30$, you'd have a pretty reasonably sized optimization problem, and in this case the images would still look pretty different. Your home for data science. computes softmin reductions on-the-fly, with a linear memory footprint: Thanks to the \(\varepsilon\)-scaling heuristic,
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