Conditional Optimal Transport for Simulation-Free Generative Modeling and Inverse Problems

Alapfogalmak

The authors develop a dynamical framework for conditional optimal transport (COT) and apply it to simulation-free conditional generative modeling and infinite-dimensional Bayesian inverse problems.

Kivonat

The content presents a study of the geometry of conditional optimal transport (COT) and proposes a simulation-free flow-based method for conditional generative modeling. The key highlights are:

The authors introduce the conditional Wasserstein space Pμ
p(Y × U) equipped with the conditional p-Wasserstein distance Wμ
p, and show that this space is a metric space that admits constant speed geodesics.

The authors characterize the absolutely continuous curves in Pμ
p(Y × U) via the continuity equation and triangular vector fields. As a corollary, they obtain a conditional version of the Benamou-Brenier theorem.

The authors propose a method for learning dynamic COT maps based on flow matching, which couples an arbitrary source distribution to a specified target distribution through a triangular COT plan.

The authors demonstrate their proposed method on two image-to-image translation tasks and an infinite-dimensional Bayesian inverse problem, showing its applicability in both finite and infinite-dimensional settings.

Statisztikák

The content does not provide any specific numerical data or metrics to support the key claims. It focuses on the theoretical development of the conditional optimal transport framework and its application to generative modeling and inverse problems.

Idézetek

"We study the geometry of conditional optimal transport (COT) and prove a dynamical formulation which generalizes the Benamou-Brenier Theorem."
"Our theory and methods are applicable in the infinite-dimensional setting, making them well suited for inverse problems."
"We build on the framework of flow matching to train a conditional generative model by approximating the geodesic path of measures induced by this COT plan."

Főbb Kivonatok

Dynamic Conditional Optimal Transport through Simulation-Free Flows

by Gavin Kerrig... : arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.04240.pdf

Mélyebb kérdések

How can the proposed conditional optimal transport framework be extended to handle more complex conditional distributions, such as multimodal or highly non-Gaussian distributions

To extend the proposed conditional optimal transport framework to handle more complex conditional distributions, such as multimodal or highly non-Gaussian distributions, several strategies can be employed. One approach is to incorporate mixture models into the framework, allowing for the representation of multiple modes in the conditional distributions. By using a mixture of conditional optimal transport plans, each corresponding to a different mode, the framework can capture the multimodal nature of the distributions. Additionally, incorporating non-Gaussian likelihood functions or priors into the optimization process can help model the complex dependencies present in the data. Techniques such as kernel density estimation or Gaussian mixture models can be used to approximate the conditional distributions in a non-Gaussian manner. By adapting the cost functions and constraints in the optimization problem to account for the specific characteristics of the conditional distributions, the framework can be tailored to handle a wide range of distributional complexities.

What are the potential limitations or challenges in applying the dynamic COT approach to high-dimensional or large-scale conditional generative modeling tasks

When applying the dynamic conditional optimal transport (COT) approach to high-dimensional or large-scale conditional generative modeling tasks, several potential limitations and challenges may arise. One significant challenge is the computational complexity associated with solving the dynamic COT problem in high-dimensional spaces. As the dimensionality of the input data increases, the optimization problem becomes more computationally demanding, requiring efficient algorithms and computational resources to handle the increased complexity. Additionally, the scalability of the framework to large-scale datasets may be limited by memory and computational constraints, necessitating the development of parallel and distributed computing strategies to handle the computational load. Furthermore, the interpretability of the results in high-dimensional spaces can be challenging, requiring advanced visualization and analysis techniques to understand the learned conditional mappings and generative models effectively.

How can the insights from the conditional optimal transport theory be leveraged to develop new algorithms or techniques for solving inverse problems in various scientific and engineering domains

The insights from the conditional optimal transport theory can be leveraged to develop new algorithms or techniques for solving inverse problems in various scientific and engineering domains. One potential application is in medical imaging, where inverse problems arise in reconstructing high-quality images from noisy or incomplete data. By formulating the inverse problem as a conditional optimal transport task, one can learn the mapping between the observed data and the desired image space, enabling more accurate and efficient image reconstruction. Additionally, in geophysical exploration, inverse problems often involve estimating subsurface properties from seismic data. By applying the principles of conditional optimal transport, researchers can develop novel approaches for solving these inverse problems, leading to improved subsurface imaging and resource exploration. Overall, the theory of conditional optimal transport offers a versatile framework for addressing inverse problems across diverse domains, providing new insights and solutions to complex data modeling challenges.