# Questions tagged [wasserstein]

The Wasserstein metric or Earth Movers Distance is a distance function between probability distributions.

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### Wasserstein distance to assess the degree of normality

The Wasserstein distance between two probability measures with quantile functions $F^{-1}$ and $G^{-1}$ is given by \begin{align} W(F,G) = \int_{[0,1]} |F^{-1}(t) - G^{-1}(t)|dt \end{align} Now let's ...
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### Upper bound for 1-Wasserstein distance between standard uniform and other distribution on $[0,1]$

I want to use the following metric to measure the distance between the standard uniform distribution and any other probability distribution on $[0,1]$. $$\int_0^1 |F(x) - x| dx$$ $F(x)$ is the cdf of ...
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I am training a WGAN to generate a vector of 6 data points (6 dimensional output). I plotted a correlation matrix of my dataset to see the underlying distribution of the 6 data points, and the ...
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### Earth Mover's Distance (EMD) between two distributions of different dimensionality

When $x_1\sim N(\mu_1, \Sigma_1)$ and $x_2 \sim N(\mu_2, \Sigma_2)$, and $\mu_1$ and $\mu_2$ are both $D$-dimensional, the EMD is well defined and there exists a closed-form solution for it. Is there ...
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### Would ratio of intragroup distances make sense for comparing two distributions?

Given two distributions a and b, we can assess the distance between them using e.g. the Wasserstein distance, Energy distance or ...
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### Closed form expression for the 2-Wasserstein distance between generalized Gaussian distributions

Essentially the title - is there a closed form expression for the 2-Wasserstein distance (aka Frechet distance, Earth Mover's distance) between two generalized Gaussian distributions? The regular ...
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### Are there any research papers which show why Wasserstein distance is better than Jensen-Shannon/KL_div/Bhattacharya distance for specific use cases?

I am trying to find reliable research work which show why displacement based metrics such as Wasserstein distance is a better suited metric than Jensen-Shannon distance in specific use cases and for ...
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### how to solve for wasserstein duality easily in a special case when 2-Wasserstein inequality constraint is binding

I was going through this nice paper ” A Simple and General Duality Proof for Wasserstein Distributionally Robust Optimization”, and one quick qu on applying Theorem 1 to my poject: What if in my ...
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### Proof for equivalence definition of convergence in the Wasserstein space

From "An Invitation to Statistics in Wasserstein Space" (Victor M. Panaretos and Yoav Zemel) Theorem 2.2.1 (Convergence in Wasserstein Space) Let $\mu, \mu_n \in \mathscr{W}_p(\mathscr{X})$. ...
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### What metrics could I use to compare multiple 2-dimensional histograms/sinograms?

I have a set of two-dimensional shapes, each represented by one or more closed paths, which enclose one continuous area. My objective is to establish a measure of similarity between these shapes. I do ...
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### How to quantify the dissimilarity across different types of variables?

I have two dataframes with the same columns but with varying sample sizes. I want to compare corresponding columns for homogeneity (i.e., do they come from the same distribution?). There are different ...
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### How to apply a seasonal index into an eCDF?

I am doing a Before/After analysis, which is aiming at evaluating the effect of a change. Assume I have made a change at the beginning of June-2022, and I want to evaluate the effect of the change ...
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### Wasserstein-1 distance between two joint distributions

Let me define random variables $A, B, C, D$ that are independent of each other. Additionally, PDFs of $A, B$ are the same, and PDFs of $C, D$ are the same too In such a case, is the following correct? ...
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### Normalized Wasserstein distance

The wasserstein_distance will be smaller the longer u_values and v_values are. ...
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### Why is the Wasserstein distance not used in Variational Inference

I just started learning the concept of variational inference in the context of variational Autoencoder, so please excuse me if the answer is obvious. I would like to know why traditionally, KL-...
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### GAN : Why does a perfect discriminator mean no gradient for the generator?

(Note : this is a cross-post from Artificial Intelligence. As I got no answer there in two weeks, I'm trying my luck on a more popuplated SE site. I know this is against SE's policy on cross-posting ...
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### How to calculate the max possible earth mover distance between histograms, given the buckets?

I have two histograms and am able to calculate the EMD / Wasserstein metric between them using the algorithm described here. In order to better communicate the implication of this metric, I want to ...
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### Earth Mover (Wasserstein) distance for ordinal discrete data

I am doing data analysis for my Masters research and which includes some Likert scale type questions. I have been calculating some distances between the responses for these questions. All this has ...
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### Is there an R or python package to calculate wasserstein metric between negative binomial distributions?

As the title says I am looking for an R or python package which can calculate wasserstein distance (aka earthmovers distance between) two lists (vectors) of sampled values from a negative binomial ...
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### Transform Earth mover's distance/Wasserstein metric to percentance (0 to 1 or 0% to 100%) [duplicate]

Ok, I do some calculations between binned histograms via the use of Wasserstein metric and more specifically this python library.. https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats....
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### Can I adjust the Wasserstein GAN loss function for my particular data?

I am working on building Generative Adversarial Networks for the purpose of generating synthetic flight data. The GAN will be trained on actual time-series flight data in the form of a (n,m,9) array ...
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### Wasserstein distance between multivariate lognormal distributions

Wikipedia gives the following formula for normal distributions: What changes, if any, do I need to make to handle multivariate lognormal distributions instead?
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### I need help understanding the meaning of the loss values of a WGAN with Gradient Penalty

I am currently working on training a Auxiliary Classifier Wasserstein GAN with Gradient Penalty. I based my implementation off of https://keras.io/examples/generative/wgan_gp/ (to which I added the ...
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### Wasserstein distance and Kolmogorov-Smirnov statistic as measures of effect size

So I've been dealing with 2 sample hypothesis testing with very large samples (around 20,000s each). Whenever I test for the equality of distribution I always reject the null hypothesis, even though ...
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### Measure for evaluating a density estimation procedure

Given an implementation of a multivariate density estimation scheme, what would be a suitable measure to evaluate the accuracy of the procedure? I am currently evaluating the procedure using three ...
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### Interpretation of upper bound on the Wasserstein Distance

I am trying to interpret the 2-Wasserstein distance and the upper bound on it. Let's say I have 2-Wasserstein distance between two distributions to be $x$, and I have an upper bound on it which gives ...
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### Intuition on Wasserstein Distance

I've been trying to familiarize myself with the Wasserstein distance and saw this answer on StackExchange by @antike that at first made a lot of sense, but then it didn't (to me, of course). In the ...
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### How can one compute the "average" of a dataset of histograms that minimizes the mean Earth Mover's Distance between all data points and average?

It is my understanding that when the distance metric is euclidean distance, the mean of a dataset minimizes the average distance between all data points and the computed "mean". In the case ...
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### What is this probability distribution?

Thank you in advance for any suggestions or feedback. I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$. I am interested in finding the Wasserstein (...
1 vote
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### When is my Wasserstein GAN-GP overfitting?

I have a hard time interpreting the WGAN-GP losses attached. At which epoch is D and/or G overfitting? The quality improves a lot overtime, yet the generator loss at later epochs does not appear on ...
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### Wasserstein distance for categorical data? Relationship to TVD?

Is the Wasserstein distance applicable for categorical data? e.g. if we have the distribution of different coloured balls in two bags, ...
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### Reference for Wasserstein distance for generalized exponential family and generalized extreme value probability distributions

The Wikipedia page gives the analytical results for computing the Wasserstein metric between two multivariate normal distributions, and I thought it used to give results for several other commonly ...
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### K-Lipschitz for the Wasserstein GANs

I am trying to follow this blog for Wasserstein loss for Generative Adversarial Networks: From GAN to WGAN. Actually, I am trying to follow the logic behind the K-Lipschitz continuity. This post in ...
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### How to check whether a partitioning is more balanced?

For example consider the following two partitions that add approximately to 100 percent. Partition 1 : 41% 36% 14% 5% 2% 1% <1% <1% Partition 2 : 82% 10% 4% 3% <1% <1% <1% <1% ...
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### Horizontal vs Vertical optimal transport of probability distributions

Optimal transport has a primal and dual form. My question is: Is the primal formulation of OT intended for horizontal movement of probability mass, whereas the dual formulation is more geared towards ...
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### Does Wasserstein distance require the source and target distributions to have the same mass?

If we minimize the Wasserstein loss, $$W_1 (P_S, P_T) = \underset{\gamma \in \Pi}{\text{min }} \sum_{x,y} |x-y|\gamma(x,y)$$ which means we are looking for the coupling $\gamma(x,y)$ that minimizes ...
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### Properties of Sinkhorn distance

I am reading the paper by Cuturi http://www.marcocuturi.net/Papers/cuturi13sinkhorn.pdf and I am curious about the properties of Sinkhorn distance and wondering what properties of the Sinkhorn ...
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### What is the intuitive difference between Wasserstein-1 distance and Wasserstein-2 distance?

What is the intuitive difference between Wasserstein-1 distance and Wasserstein-2 distance, and how to know which one to use?
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### What are the advantages of Wasserstein distance compared to Jensen-Shannon divergence?

What is the practical difference between Wasserstein metric and Jensen-Shannon divergence? Wasserstein metric is also referred to as Earth mover's distance. From Wikipedia: Wasserstein metric is a ...
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### How to Show Wasserstein Metric is Sum Invariant?

A paper I'm trying to understand states that that Wasserstein metric obeys certain properties, which I'd like to prove. This metric is defined for two random variables $U, V$ and $p \in (1, \infty)$ ...
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### Multivariate Wasserstein metric for $n$-dimensions

I am a vegetation ecologist and poor student of computer science who recently learned of the Wasserstein metric. Application of this metric to 1d distributions I find fairly intuitive, and inspection ...
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### Optimal critic in Wasserstein GAN (WGAN)

In the Wasserstein GAN (generative adversarial network) paper they say that the critic needs to be optimal. What does this mean practically? They do more training iterations of critic (5) than the ...
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### Weight Clipping range with WGAN and relation with other factors

For a while, my code with WGAN has failed to generate quality images through a complicated multi-class database. My issue has been with implementation and not code. Recently I read the WGAN-GP paper ...
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### Relation Between Wasserstein Distance and KL-Divergence (Relative Entropy)

Consider the Wasserstein metric of order one $W_1$ (a.k.a. the Earth Movers Distance). I would like to know whether it is possible to link $W_1$ and Kullback–Leibler divergence (a.k.a. relative ...
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