🤖 AI Summary
Conditional Flow Matching (CFM) suffers from high variance in vector field estimation and limited generation quality when applied to highly correlated data such as time series. To address this, we propose an implicit stochastic path-based conditional probability flow modeling framework grounded in Bayesian decision theory. Our approach is the first to integrate Gaussian processes (GPs) into the CFM paradigm, leveraging the key property that the derivative of a GP remains a GP—enabling analytical, simulation-free sampling. The method supports joint multi-point modeling and principled incorporation of prior knowledge. Empirical evaluation demonstrates substantial reduction in marginal vector field estimation variance. On benchmark datasets including handwritten digit generation, our method achieves significant improvements in FID and LPIPS scores, confirming concurrent enhancement in both sample fidelity and generalization capability.
📝 Abstract
Flow matching (FM) is a family of training algorithms for fitting continuous normalizing flows (CNFs). A standard approach to FM, called conditional flow matching (CFM), exploits the fact that the marginal vector field of a CNF can be learned by fitting least-square regression to the so-called conditional vector field specified given one or both ends of the flow path. We show that viewing CFM training from a Bayesian decision theoretic perspective on parameter estimation opens the door to generalizations of CFM algorithms. We propose one such extension by introducing a CFM algorithm based on defining conditional probability paths given what we refer to as ``streams'', instances of latent stochastic paths that connect pairs of noise and observed data. Further, we advocate the modeling of these latent streams using Gaussian processes (GPs). The unique distributional properties of GPs, and in particular the fact that the velocity of a GP is still a GP, allows drawing samples from the resulting stream-augmented conditional probability path without simulating the actual streams, and hence the ``simulation-free"nature of CFM training is preserved. We show that this generalization of the CFM can substantially reduce the variance in the estimated marginal vector field at a moderate computational cost, thereby improving the quality of the generated samples under common metrics. Additionally, we show that adopting the GP on the streams allows for flexibly linking multiple related training data points (e.g., time series) and incorporating additional prior information. We empirically validate our claim through both simulations and applications to two hand-written image datasets.