Monge Map

Monge maps, optimal transport mappings that minimize the cost of transforming one probability distribution into another, are a central focus in current optimal transport research. Recent work emphasizes learning these maps using neural networks, particularly focusing on extending their applicability to "incomparable spaces" (e.g., different dimensions) and incorporating cost functions tailored to specific applications, such as disentangled representation learning and domain adaptation. These advancements are improving the efficiency and interpretability of Monge map estimation, with significant implications for diverse fields including single-cell biology, image translation, and fair machine learning. The development of novel regularizers and algorithms is driving progress towards more robust and computationally efficient methods.