Stanford CME296 Diffusion & Large Vision Models | Spring 2026 | Lecture 3 - Flow matching

Key concepts include the initial distribution (P0, usually noise), target distribution (P1, clean data), trajectories (XT, the path taken by an observation), flow (S(t, x0), a function mapping initial conditions to points at time t), probability path (P_t(X), the distribution of observations at time t), and the central vector field (U_t(X), indicating direction and speed of particle movement).

The vector field (velocity) in flow matching provides direct instructions for how particles should move and at what speed. The score, in score matching, acts more like a compass, indicating regions of high density without specifying a direct movement vector.

Lipschitz continuity is a mathematical property that ensures that if the learned vector field satisfies it, then the trajectories generated by following that field will be unique for any given initial condition. This guarantees a one-to-one mapping between the velocity field and the flow, ensuring consistent particle paths.

The continuity equation expresses the conservation of mass within a probability distribution, stating that the temporal evolution of density in a region is determined by the net inflow minus outflow of density. It provides a macroscopic view of how the probability path evolves, linking the vector field to the probability distribution.

Earlier methods like Continuous Normalizing Flows tried to learn the vector field by maximizing likelihood, which required constantly solving expensive numerical integrals during training. This made the approach very slow and impractical, leading to the development of flow matching's direct learning approach.

Instead of directly transporting from the initial distribution to the complex target data distribution, flow matching first formulates a simpler problem: transporting the initial distribution to a single point (Dirac Delta distribution). This allows for a tractable conditional probability path and vector field.

The CFM loss is an extremely tractable loss function derived from the marginal vector field. It allows the model to directly learn the desired velocity field by minimizing the squared distance between the learned velocity and a simple target (x1 - x0), making optimization efficient and effective.

During training, the model samples noise (x0), clean images (x1), and time steps, constructs a noisy image (Xt), and learns the vector field by predicting x1 - x0 using the CFM loss. For inference, it samples from a standard normal distribution and numerically solves the ODE by following the learned vector field step-by-step to generate the final image.

The Reflow procedure is a fine-tuning technique that addresses the issue of learned trajectories becoming curved rather than straight. It involves using the newly generated pairs (from integrating the learned ODE) as new 'straight lines' to retrain the model, making trajectories straighter and enabling faster inference with fewer steps.

While Reflow helps straighten trajectories, it introduces discretization errors from numerical integration and approximation errors from the neural network. Performing it too many times can lead to quality degradation. It trades off simplicity and potentially lower quality for significantly faster inference speeds.

These three generative frameworks are different lenses to solve the same problem. They can be unified under a broader theory (like stochastic interpolants) which shows that if you know any two of noise, score, or velocity, you can deduce the third, highlighting their deep interconnections.