Flow Matching & Optimal Transport

Continuous Flow Matching (Lipman et al.) & Optimal Transport Flow Matching (OT-FM, powering Stable Diffusion 3 and Flux.1) revolutionizes continuous generative modeling by replacing curved stochastic diffusion SDE paths with mathematically minimal straight-line probability trajectories; standard diffusion processes follow curved, chaotic Brownian paths requiring fine-grained numerical integration steps; Flow Matching directly regresses the time-dependent vector field $v_t(x)$ generating continuous probability path $\psi_t(x_0)$: $\mathcal{L}_{\text{CFM}}(\theta) = \mathbb{E}[ \|v_\theta(\psi_t(x_0), t) - \dot{\psi}_t(x_0)\|^2 ]$; under Optimal Transport conditions, the interpolation trajectory from noise $x_0 \sim \mathcal{N}(0, I)$ to data $x_1$ becomes a pure straight line: $\psi_t(x_0) = (1 - t) x_0 + t x_1$, where the target vector velocity field is the constant vector $\dot{\psi}_t = x_1 - x_0$, enabling 4-10 step ultra-fast generation via simple Euler ODE steps.

Next-generation state-of-the-art visual generation architectures (Flux.1, SD3), fast continuous normalizing flows, and multi-modal trajectory synthesis.

Eliminates trajectory curvature and variance in stochastic diffusion, achieving state-of-the-art image synthesis in single-digit ODE integration steps.

Derive the Conditional Flow Matching (CFM) loss showing why optimal transport linear paths yield constant velocity targets $\dot{\psi}_t = x_1 - x_0$?

Explain why Monge-Kantorovich Optimal Transport couplings minimize total kinetic transportation energy across trajectories?

Explain the Multimodal Diffusion Transformer (MMDiT) architecture predicting Flow Matching vector fields across joint vision-text streams?

Explain how Rectified Flow's Re-Flow procedure straightens trajectories further to enable 1-step or 2-step ultra-fast distillation?

Derive exact log-likelihood computation in Flow Matching using the continuous instantaneous change of variables formula and Hutchinson's trace estimator?