Small Toy-Example of using the Kantorovich-Rubinstein duality of the Wasserstein-1 metric to estimate the distance between two 1D distributions. According to this duality, the Wasserstein-1 distance between two distributions P and Q is:

$$W_1(P||Q) = \underset{\lVert h \rVert_L \leq 1}{\text{sup}} \underset{x \sim P}{\mathbb{E}}[h(x)] - \underset{x \sim Q}{\mathbb{E}}[h(x)],$$

where the supremum is taken over all 1-Lipschitz functions. We estimate $h$ using a fully-connected neural network. To enforce the Lipschitz constraint, we use projected gradient descent proposed in 'Regularisation of Neural Networks by Enforcing Lipschitz Continuity' which divides each weight matrix by its spectral norm if it surpasses 1. As the Lipschitz constant of the composition of functions is smaller than the product of their individual Lipschitz constants, this ensures that our entire network is 1-Lipschitz. The same duality is also the main idea behind Wasserstein GANs, where the discriminator has the same interpretation as our function $h$.

To get an idea about the accuracy of our estimation, we can use the special formula for the $W_1$ distance between two one-dimensional distributions with CDFs $F$ and $G$:

$$ W_1(P||Q) = \int_\mathbb{R} |F(x) - G(x) |dx.$$

To evaluate this expression, we use simple numerical integration.