Efficient Approximation of Wasserstein Distances and Sobolev-Smooth Functions on Probability Spaces using Machine Learning Techniques

Concepts de base

This work presents three machine learning-based approaches to efficiently approximate Wasserstein distances and other Sobolev-smooth functions defined on probability spaces: (1) solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials, (2) employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces, and (3) addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. The authors provide explicit and quantitative bounds on generalization errors for each of these solutions, leveraging the theory of metric Sobolev spaces, optimal transport, variational calculus, and large deviation bounds.

Résumé

The paper focuses on the efficient numerical approximation of Sobolev-smooth functions defined on spaces of probability measures. The authors use the Wasserstein distance as a motivating example, but the techniques developed can be applied to a broader class of Wasserstein Sobolev functions.
The key contributions are:

A constructive approach to approximate the Wasserstein distance using a finite number of Kantorovich potentials, with quantitative convergence rates based on random subcoverings of the probability space.

An empirical risk minimization framework with Tikhonov regularization in Wasserstein Sobolev spaces, which provides deterministic convergence results as well as explicit generalization error bounds in the presence of noisy data.

A saddle point formulation to solve the Euler-Lagrange equation of the Tikhonov functional, which is addressed using an adversarial deep neural network approach.

The theoretical results are complemented by numerical experiments on the MNIST and CIFAR-10 datasets, demonstrating the efficiency and accuracy of the proposed methods compared to state-of-the-art techniques for computing the Wasserstein distance.

Stats

The Wasserstein distance between two probability measures µ and ν is defined as:
Wp(µ, ν) = (inf_γ ∫_U×U ρ(x, y)^p dγ(x, y))^(1/p)
where Γ(µ, ν) denotes the set of Borel probability measures on U × U having µ and ν as marginals.
The p-th moment of a probability measure µ ∈ P(U) is defined as:
mp,x0(µ) = (∫_U ρ(x, x0)^p dµ(x))^(1/p)
where x0 ∈ U is a fixed point.

Citations

"The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable."
"In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches."

Idées clés tirées de

Approximation Theory, Computing, and Deep Learning on the Wasserstein Space

by Massimo Forn... à arxiv.org 05-01-2024

https://arxiv.org/pdf/2310.19548.pdf

Questions plus approfondies

How can the proposed techniques be extended to approximate other types of functions defined on probability spaces beyond the Wasserstein distance

The techniques proposed in the study can be extended to approximate other types of functions defined on probability spaces beyond the Wasserstein distance by leveraging the concept of Sobolev spaces and empirical risk minimization. These methods can be applied to approximate Sobolev-smooth functions on metric measure spaces by utilizing neural networks as basis functions. By defining suitable functionals and employing regularization techniques, it is possible to approximate a wide range of functions beyond just the Wasserstein distance. The key lies in adapting the approach to the specific function being approximated and ensuring that the chosen neural network architecture is capable of capturing the essential features of the function.

What are the limitations of the adversarial deep neural network approach for solving the saddle point problem, and how can they be addressed

The adversarial deep neural network approach for solving the saddle point problem has certain limitations that need to be addressed. One limitation is the potential for instability in training the adversarial networks, leading to convergence issues and suboptimal solutions. This can be mitigated by carefully tuning hyperparameters, adjusting the network architecture, and implementing regularization techniques to improve stability during training. Additionally, the computational complexity of the adversarial training process can be high, requiring efficient optimization algorithms and computational resources. To address this, techniques such as mini-batch training, gradient clipping, and early stopping can be employed to enhance the training process and improve convergence.

Can the theoretical results on generalization error bounds be further improved by incorporating more advanced techniques from the field of deep learning

The theoretical results on generalization error bounds can be further improved by incorporating more advanced techniques from the field of deep learning. One approach is to explore the use of advanced regularization methods such as dropout, batch normalization, and weight decay to prevent overfitting and improve the generalization capabilities of the neural networks. Additionally, techniques like transfer learning, ensemble learning, and adversarial training can be utilized to enhance the robustness and performance of the models. By integrating these advanced deep learning techniques into the training process, it is possible to achieve better generalization error bounds and improve the overall accuracy and reliability of the approximations.