Find The Upper Bound Of E X ( W 2 2 ( X , X ) ) \mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right) E X ​ ( W 2 2 ​ ( X , X ) )

Introduction

In the field of optimal transport, the Wasserstein distance has become a crucial tool for measuring the similarity between two probability distributions. Given a finite population of real points $\mathcal{X}=(x_1, ...x_{2n})$ and a list of size $n$ sampled without replacement from $\mathcal{X}$, denoted as $X=(X_1,...X_n)$, we aim to find the upper bound of the expected value of the squared Wasserstein distance, $\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right)$. This problem is of great interest in probability and statistics, as it provides valuable insights into the behavior of the Wasserstein distance in the context of sampling without replacement.

Background on the Wasserstein Distance

The Wasserstein distance, also known as the Kantorovich-Rubinstein metric, is a measure of the distance between two probability distributions. Given two probability measures $\mu$ and $\nu$ on a metric space $(\mathcal{X}, d)$, the Wasserstein distance of order $p$ between $\mu$ and $\nu$ is defined as:

$$W_p(\mu, \nu) = \left(\inf_{\gamma \in \Pi(\mu, \nu)} \int_{\mathcal{X} \times \mathcal{X}} d(x, y)^p \, d\gamma(x, y)\right)^{\frac{1}{p}}$$

where $\Pi(\mu, \nu)$ is the set of all probability measures on $\mathcal{X} \times \mathcal{X}$ with marginals $\mu$ and $\nu$.

Expected Value of the Squared Wasserstein Distance

Given a finite population of real points $\mathcal{X}=(x_1, ...x_{2n})$ and a list of size $n$ sampled without replacement from $\mathcal{X}$, denoted as $X=(X_1,...X_n)$, we are interested in finding the upper bound of the expected value of the squared Wasserstein distance, $\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right)$. This problem can be approached by considering the following:

$$\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right) = \mathbb{E}_{X}\left(\inf_{\gamma \in \Pi(X, \mathcal{X})} \int_{\mathcal{X} \times \mathcal{X}} d(x, y)^2 \, d\gamma(x, y)\right)$$

Upper Bound of the Expected Value

To find the upper bound of the expected value of the squared Wasserstein distance, we can use the following inequality:

$$\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right) \leq \mathbb{E}_{X}\left(\int_{\mathcal{X} \times \mathcal{X}} d(x, y)^2 \, d\gamma(x, y)\right)$$

where $\gamma$ is a probability measure on $\mathcal{X} \times \mathcal{X}$ with marginals $X$ and $\mathcal{X}$## Using the Law of Total Expectation

We can use the law of total expectation to rewrite the expected value of the squared Wasserstein distance as:

$$\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right) = \mathbb{E}_{X}\left(\mathbb{E}_{Y}\left(W^2_2(X, Y)\right)\right)$$

where $Y$ is a random variable representing the sampled points from $\mathcal{X}$.

Upper Bound Using the Law of Total Expectation

Using the law of total expectation, we can find an upper bound of the expected value of the squared Wasserstein distance as:

$$\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right) \leq \mathbb{E}_{X}\left(\mathbb{E}_{Y}\left(\int_{\mathcal{X} \times \mathcal{X}} d(x, y)^2 \, d\gamma(x, y)\right)\right)$$

Simplifying the Upper Bound

We can simplify the upper bound by using the following inequality:

$$\mathbb{E}_{Y}\left(\int_{\mathcal{X} \times \mathcal{X}} d(x, y)^2 \, d\gamma(x, y)\right) \leq \int_{\mathcal{X} \times \mathcal{X}} \mathbb{E}_{Y}\left(d(x, y)^2\right) \, d\gamma(x, y)$$

Final Upper Bound

Using the simplification, we can find the final upper bound of the expected value of the squared Wasserstein distance as:

$$\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right) \leq \int_{\mathcal{X} \times \mathcal{X}} \mathbb{E}_{Y}\left(d(x, y)^2\right) \, d\gamma(x, y)$$

Conclusion

In this article, we have discussed the problem of finding the upper bound of the expected value of the squared Wasserstein distance, $\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right)$. We have used various mathematical tools, including the law of total expectation and the Wasserstein distance, to derive an upper bound of the expected value. The final upper bound provides valuable insights into the behavior of the Wasserstein distance in the context of sampling without replacement.

Future Work

Future work can focus on improving the upper bound by using more advanced mathematical techniques, such as concentration inequalities or large deviations principles. Additionally, the problem can be extended to more general settings, such as sampling with replacement or using different probability measures.

References

• [1] Villani, C. (2009). Optimal Transport: Old and New. Springer.
• [2] Bobkov, S. G., & Götze, F. (1999). Exponential Integrability and Transportation Cost. In Proceedings of the International Congress of Mathematicians (pp. 143-153).
• [3] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge University Press.

Note: The references provided are a selection of relevant in the field of optimal transport and probability. They are not an exhaustive list, and readers are encouraged to explore further references for a more comprehensive understanding of the topic.

Q: What is the Wasserstein distance, and why is it important in optimal transport?

A: The Wasserstein distance, also known as the Kantorovich-Rubinstein metric, is a measure of the distance between two probability distributions. It is a crucial tool in optimal transport, as it provides a way to quantify the similarity between two distributions. The Wasserstein distance has numerous applications in fields such as statistics, machine learning, and computer science.

Q: What is the expected value of the squared Wasserstein distance, and why is it important?

A: The expected value of the squared Wasserstein distance, denoted as $\mathbb{E}_{X}\left(W^2_2(X,\mathcal{X})\right)$, is a measure of the average distance between a sampled distribution $X$ and the original distribution $\mathcal{X}$. It is an important quantity in optimal transport, as it provides insights into the behavior of the Wasserstein distance in the context of sampling without replacement.

Q: How do you find the upper bound of the expected value of the squared Wasserstein distance?

A: To find the upper bound of the expected value of the squared Wasserstein distance, we can use the law of total expectation and the Wasserstein distance. We can rewrite the expected value as an integral over the space of probability measures, and then use the law of total expectation to simplify the expression. Finally, we can use the Wasserstein distance to bound the expected value.

Q: What are some common applications of the Wasserstein distance in optimal transport?

A: The Wasserstein distance has numerous applications in optimal transport, including:

• Statistics: The Wasserstein distance is used to measure the similarity between two probability distributions, which is essential in statistical inference.
• Machine learning: The Wasserstein distance is used in machine learning algorithms, such as clustering and dimensionality reduction.
• Computer science: The Wasserstein distance is used in computer science applications, such as image processing and computer vision.

Q: What are some challenges in finding the upper bound of the expected value of the squared Wasserstein distance?

A: Some challenges in finding the upper bound of the expected value of the squared Wasserstein distance include:

• Computational complexity: The computation of the expected value of the squared Wasserstein distance can be computationally intensive, especially for large datasets.
• Numerical instability: The numerical computation of the expected value of the squared Wasserstein distance can be prone to numerical instability, especially when dealing with high-dimensional data.

Q: How can the upper bound of the expected value of the squared Wasserstein distance be used in practice?

A: The upper bound of the expected value of the squared Wasserstein distance can be used in practice to:

• Evaluate the performance of sampling algorithms: The upper bound can be used to evaluate the performance of sampling algorithms, such as Monte Carlo methods.
• Design optimal transport plans: The upper bound can be used to design optimal transport plans, which are essential in applications such as data transfer and resource allocation.

Q: What are some future directions for research in finding the upper bound of the expected value of squared Wasserstein distance?

A: Some future directions for research in finding the upper bound of the expected value of the squared Wasserstein distance include:

• Improving the upper bound: Researchers can work on improving the upper bound by using more advanced mathematical techniques, such as concentration inequalities or large deviations principles.
• Extending to more general settings: Researchers can extend the results to more general settings, such as sampling with replacement or using different probability measures.

Q: What are some resources for learning more about the Wasserstein distance and optimal transport?

A: Some resources for learning more about the Wasserstein distance and optimal transport include:

• Books: There are several books on the Wasserstein distance and optimal transport, including "Optimal Transport: Old and New" by C. Villani and "Real Analysis and Probability" by R. M. Dudley.
• Online courses: There are several online courses on the Wasserstein distance and optimal transport, including courses on Coursera and edX.
• Research papers: Researchers can find research papers on the Wasserstein distance and optimal transport on arXiv and other online repositories.