Optimal Transport (OT) Theory: A New Model for Transport Maps
Optimal Transport (OT) theory is a powerful concept that focuses on finding the most efficient maps to morph one probability measure into another. These maps, known as “thriftiest” maps, aim to minimize the average cost between the original measure and its transformed image. In this article, we will explore a new model for transport maps and its significance in the field of Artificial Intelligence (AI).
Traditionally, many computational approaches have been proposed to estimate these “thriftiest” maps. Some methods utilize entropic maps or neural networks to calculate the optimal transport. However, we propose a different approach using a family of translation invariant costs, incorporating a regularizer. This model offers a broader scope and can be applied to various scenarios.
We introduce a generalization of the entropic map specifically designed for our model. Surprisingly, we discovered a significant correlation between this map and the Bregman centroids of the divergence generated by the regularizer, as well as the proximal operator. This connection enhances the efficiency and accuracy of our approach.
By selecting a sparsity-inducing norm, our model generates transport maps that adhere to Occam’s razor principle. Occam’s razor suggests that the simplest explanation is often the best. In our case, the displacement vectors induced by our maps are sparse, meaning they prioritize minimal movement. The sparsity pattern of these vectors varies depending on the chosen norm, allowing for flexibility and adaptability in different scenarios.
To demonstrate the effectiveness of our method, we applied it to high-dimensional single-cell transcription data. This data represents gene counts for individual cells without undergoing dimensionality reduction. As a result, our approach retains the ability to interpret all displacements at the gene level, providing valuable insights in the field of gene expression analysis.
In conclusion, our new model for transport maps offers a promising solution within the field of AI and optimal transport theory. Its translation invariant costs, connection to Bregman centroids, and sparsity-inducing norm result in efficient and interpretable transport maps. By eliminating the need for dimensionality reduction, our method enables comprehensive analysis of high-dimensional data. It’s a groundbreaking approach that opens doors to new possibilities in AI research.