Conditional vs Marginal Paths

Interactive · Conditional Paths vs ODE Trajectories

Data lives at two points x₀ ∈ {−μ, +μ}; noise is ε ~ 𝒩(0,1). Each gray line is a conditional path xₜ = (1−t)x₀ + tε — always perfectly straight. Each red curve is an ODE trajectory following the marginal velocity v(x,t) = 𝔼[ε−x₀ | xₜ=x] — clearly curved, because the velocity averages over all possible (x₀, ε) pairs. Click on the right edge (t=1) to launch new ODE trajectories.

Legend

conditional path (straight)

ODE trajectory (curved!)

marginal velocity field

Data separation μ = 2.5

conditional paths velocity field ODE trajectories

t=0 (left) = data
t=1 (right) = noise
ODE goes right→left (sampling)

The conditional paths are all straight — by construction. But the ODE trajectories that a sampler actually follows are curved, because the marginal velocity field routes particles nonlinearly toward data modes. This is why linear interpolation ≠ fast sampling: the solver must still trace these curves accurately.