# What is the advantages of Wasserstein metric compared to Kullback-Leibler divergence?

What is the practical difference between Wasserstein metric and Kullback-Leibler divergence? Wasserstein metric is also referred to as Earth mover's distance.

From Wikipedia:

Wasserstein (or Vaserstein) metric is a distance function defined between probability distributions on a given metric space M.

and

Kullback–Leibler divergence is a measure of how one probability distribution diverges from a second expected probability distribution.

I've seen KL been used in machine learning implementations, but I recently came across the Wasserstein metric. Is there a good guideline on when to use one or the other?

(I have insufficient reputation to create a new tag with Wasserstein or Earth mover's distance.)

When considering the advantages of Wasserstein metric compared to KL divergence, then the most obvious one is that W is a metric whereas KL divergence is not, since KL is not symmetric (i.e. $D_{KL}(P||Q) \neq D_{KL}(Q||P)$ in general) and does not satisfy the triangle inequality (i.e. $D_{KL}(R||P) \leq D_{KL}(Q||P) + D_{KL}(R||Q)$ does not hold in general).

As what comes to practical difference, then one of the most important is that unlike KL (and many other measures) Wasserstein takes into account the metric space and what this means in less abstract terms is perhaps best explained by an example (feel free to skip to the figure, code just for producing it):

# define samples this way as scipy.stats.wasserstein_distance can't take probability distributions directly
sampP = [1,1,1,1,1,1,2,3,4,5]
sampQ = [1,2,3,4,5,5,5,5,5,5]
# and for scipy.stats.entropy (gives KL divergence here) we want distributions
P = np.unique(sampP, return_counts=True)[1] / len(sampP)
Q = np.unique(sampQ, return_counts=True)[1] / len(sampQ)
# compare to this sample / distribution:
sampQ2 = [1,2,2,2,2,2,2,3,4,5]
Q2 = np.unique(sampQ2, return_counts=True)[1] / len(sampQ2)

fig = plt.figure(figsize=(10,7))
plt.subplot(2,2,1)
plt.bar(np.arange(len(P)), P, color='r')
plt.xticks(np.arange(len(P)), np.arange(1,5), fontsize=0)
plt.subplot(2,2,3)
plt.bar(np.arange(len(Q)), Q, color='b')
plt.xticks(np.arange(len(Q)), np.arange(1,5))
plt.title("Wasserstein distance {:.4}\nKL divergence {:.4}".format(
scipy.stats.wasserstein_distance(sampP, sampQ), scipy.stats.entropy(P, Q)), fontsize=10)
plt.subplot(2,2,2)
plt.bar(np.arange(len(P)), P, color='r')
plt.xticks(np.arange(len(P)), np.arange(1,5), fontsize=0)
plt.subplot(2,2,4)
plt.bar(np.arange(len(Q2)), Q2, color='b')
plt.xticks(np.arange(len(Q2)), np.arange(1,5))
plt.title("Wasserstein distance {:.4}\nKL divergence {:.4}".format(
scipy.stats.wasserstein_distance(sampP, sampQ2), scipy.stats.entropy(P, Q2)), fontsize=10)
plt.show()


Here the measures between red and blue distributions are the same for KL divergence whereas Wasserstein distance measures the work required to transport the probability mass from the red state to the blue state using x-axis as a “road”. This measure is obviously the larger the further away the probability mass is (hence the alias earth mover's distance). So which one you want to use depends on your application area and what you want to measure. As a note, instead of KL divergence there are also other options like Jensen-Shannon distance that are proper metrics.

Wasserstein metric most commonly appears in optimal transport problems where the goal is to move things from a given configuration to a desired configuration in the minimum cost or minimum distance. The Kullback-Leibler (KL) is a divergence (not a metric) and shows up very often in statistics, machine learning, and information theory.

Also, the Wasserstein metric does not require both measures to be on the same probability space, whereas KL divergence requires both measures to be defined on the same probability space.

Perhaps the easiest spot to see the difference between Wasserstein distance and KL divergence is in the multivariate Gaussian case where both have closed form solutions. Let's assume that these distributions have dimension $$k$$, means $$\mu_i$$, and covariance matrices $$\Sigma_i$$, for $$i=1,2$$. They two formulae are:

$$W_{2} (\mathcal{N}_0, \mathcal{N}_1)^2 = \| \mu_1 - \mu_2 \|_2^2 + \mathop{\mathrm{tr}} \bigl( \Sigma_1 + \Sigma_2 - 2 \bigl( \Sigma_2^{1/2} \Sigma_1 \Sigma_2^{1/2} \bigr)^{1/2} \bigr)$$ and $$D_\text{KL} (\mathcal{N}_0, \mathcal{N}_1) = \frac{1}{2}\left( \operatorname{tr} \left(\Sigma_1^{-1}\Sigma_0\right) + (\mu_1 - \mu_0)^\mathsf{T} \Sigma_1^{-1}(\mu_1 - \mu_0) - k + \ln \left(\frac{\det\Sigma_1}{\det\Sigma_0}\right) \right).$$ To simplify let's consider $$\Sigma_1=\Sigma_2=wI_k$$ and $$\mu_1\neq\mu_2$$. With these simplifying assumptions the trace term in Wasserstein is $$0$$ and the trace term in the KL divergence will be 0 when combined with the $$-k$$ term and the log-determinant ratio is also $$0$$, so these two quantities become: $$W_{2} (\mathcal{N}_0, \mathcal{N}_1)^2 = \| \mu_1 - \mu_2 \|_2^2$$ and $$D_\text{KL} (\mathcal{N}_0, \mathcal{N}_1) = (\mu_1 - \mu_0)^\mathsf{T} \Sigma_1^{-1}(\mu_1 - \mu_0).$$ Notice that Wasserstein distance does not change if the variance changes (say take $$w$$ as a large quantity in the covariance matrices) whereas the KL divergence does. This is because the Wasserstein distance is a distance function in the joint support spaces of the two probability measures. In contrast the KL divergence is a divergence and this divergence changes based on the information space (signal to noise ratio) of the distributions.

The Wasserstein metric is useful in validation of models as its units are that of the response itself. For example, if you are comparing two stochastic representations of the same system (e.g. a reduced-order-model), $$P$$ and $$Q$$, and the response is units of displacement, the Wasserstein metric is also in units of displacement. If you were reduce your stochastic representation to a deterministic, the distribution's CDF of each is a step function. The Wasserstein metric is the difference of the values.

I find this property to be a very natural extension to talk about the absolute difference between two random variables