A Distributed Framework for the Construction of Transport Maps.

Area of Science:

• Machine Learning
• Computational Statistics
• Applied Mathematics

Background:

• Reasoning under uncertainty in large, complex datasets is crucial.
• Transforming probability distributions is key for Bayesian inference and generative modeling.
• High-dimensional transformations present significant computational challenges.

Purpose of the Study:

• To develop efficient and parallelizable methods for computing measure transport maps.
• To address computational difficulties in transforming probability distributions.
• To enable scalable learning of transport maps for complex data.

Main Methods:

• Formulating measure transport as a convex optimization problem via relative entropy minimization.
• Employing empirical minimization and polynomial chaos map parameterization.
• Leveraging nonequilibrium thermodynamics for sequential learning of composite transport maps.

Main Results:

• A convex optimization framework for learning transport maps under mild assumptions.
• Demonstrated scalability and parallelizability of the proposed methods.
• Successful application to Bayesian inference (Boston housing) and generative modeling (MNIST).

Conclusions:

• The proposed framework offers an efficient and scalable solution for learning measure transport maps.
• The methods are applicable to diverse scientific problems involving probability distribution transformations.
• This work advances computational approaches for uncertainty quantification in complex data.