Optimal Transport Theory and Applications
Summary
Optimal transport theory provides a rigorous framework for comparing and transforming probability distributions by minimising a cost function associated with “moving” mass from one configuration to another. Originating in the eighteenth century with Monge’s formulation and later generalised by Kantorovich to allow mass splitting, the theory has evolved into a versatile tool across mathematics, physics and data science. Central constructs include the family of Wasserstein distances, which quantify discrepancies between measures, and dual formulations that link to convex analysis. Recent advances have focused on scalable algorithms, entropy-regularised formulations and extensions to unbalanced transport. Applications range from image processing and machine learning to fluid dynamics, economics and network analysis, reflecting the broad impact of optimal transport in modelling and inference problems on large and structured data.
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Optimal Transport Theory and Applications publication trend
The graph below shows the total number of articles in optimal transport theory and applications across all publications each year (not limited to Nature Index journals).
Technical terms
Optimal Transport (OT): A mathematical framework for finding the least‐cost plan to move one measure to another under a given cost function.
Wasserstein distance: A family of metrics on probability distributions defined by the minimal transport cost raised to a power, capturing spatial discrepancies.
Transportation Lp (TLp) distance: A generalisation of Wasserstein metrics that treats signals as functions rather than measures, enabling direct application to multichannel data without mass constraints.
Hellinger–Kantorovich distance: A hybrid metric combining transport cost with entropy penalisation, allowing comparison of measures with different total mass and interpolating between transport and Hellinger divergences.
References
- Scalable Optimal Transport Methods in Machine Learning: A Contemporary Survey. IEEE Transactions on Pattern Analysis and Machine Intelligence (2024).
- A linear transportation L p distance for pattern recognition. Pattern Recognition (2024).
- Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones Mathematicae (2017).
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