Carré du champ flow matching: better quality-generalisation tradeoff in generative models
This work addresses a fundamental problem in generative modeling for AI applications, particularly in science where data is often limited or unevenly distributed, though it is incremental as it builds upon existing flow matching methods.
The paper tackled the tradeoff between sample quality and memorization in deep generative models by introducing Carré du champ flow matching (CDC-FM), which uses geometry-aware noise to regularize probability paths, resulting in improved quality-generalization tradeoffs, especially in data-scarce and non-uniformly sampled datasets.
Deep generative models often face a fundamental tradeoff: high sample quality can come at the cost of memorisation, where the model reproduces training data rather than generalising across the underlying data geometry. We introduce Carré du champ flow matching (CDC-FM), a generalisation of flow matching (FM), that improves the quality-generalisation tradeoff by regularising the probability path with a geometry-aware noise. Our method replaces the homogeneous, isotropic noise in FM with a spatially varying, anisotropic Gaussian noise whose covariance captures the local geometry of the latent data manifold. We prove that this geometric noise can be optimally estimated from the data and is scalable to large data. Further, we provide an extensive experimental evaluation on diverse datasets (synthetic manifolds, point clouds, single-cell genomics, animal motion capture, and images) as well as various neural network architectures (MLPs, CNNs, and transformers). We demonstrate that CDC-FM consistently offers a better quality-generalisation tradeoff. We observe significant improvements over standard FM in data-scarce regimes and in highly non-uniformly sampled datasets, which are often encountered in AI for science applications. Our work provides a mathematical framework for studying the interplay between data geometry, generalisation and memorisation in generative models, as well as a robust and scalable algorithm that can be readily integrated into existing flow matching pipelines.