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A short introduction to optimal transport and Wasserstein distance (2020)
Short Introduction to Optimal Transport and Wasserstein Distance09 Oct 2020 These notes provide a brief introduction to optimal transport theory, prioritizing intuition over mathematical rigor. A more rigorous presentation would require some additional background in measure theory.
Rather than getting lost in these details, the important point is that we have reduced the 2D problem to something similar to the 1D example we considered in the last section, and we can use the same code to identify the optimal transport plan, \(\mathbf{T}^*\). It is pretty difficult to visually interpret this optimal transport plan as it is extremely sparse — in fact, I had to add a little bit of Gaussian blur to the heatmap so that the yellow spots, corresponding to peaks in \(\mathbf{T}^*\), are visible. You can think of the regularization term as reducing sparsity in optimal transport plan and discouraging the solution from hiding out in the sharp edges of the polytope defined by the linear constraints of the problem.
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