A Vectorial Approach to Unbalanced Optimal Mass Transport

Area of Science:

• Computational Mathematics
• Applied Mathematics
• Data Science

Background:

• Optimal Mass Transport (OMT) traditionally enforces mass conservation.
• The Benamou-Brenier formulation is a standard for OMT.
• Unbalanced OMT relaxes mass conservation by incorporating source terms.

Purpose of the Study:

• To extend the vector-valued optimal mass transport framework to handle unbalanced densities.
• To demonstrate a unified approach for both scalar and vector-valued unbalanced OMT.
• To validate the proposed method on diverse datasets.

Main Methods:

• Incorporation of an auxiliary source layer into the vector-valued OMT framework.
• Definition of the source term as the flow between the auxiliary source and original layers.
• Application of the method to scalar and vector-valued density settings.

Main Results:

• Successfully integrated unbalanced optimal mass transport into the vector-valued framework.
• Demonstrated the capability to model unbalanced densities in both scalar and vector settings.
• Validated the approach using synthetic and real-world vector-valued datasets.

Conclusions:

• The proposed method provides a unified and effective way to handle unbalanced optimal mass transport.
• The addition of a source layer simplifies the integration of unbalanced models.
• The framework is versatile and applicable to various data types.