Rectified Flow

Given any coupling of two distributions — in the generative case $X_0 \sim \pi_0$ noise and $X_1 \sim \pi_1$ data, drawn independently — the rectified flow induced by that pair is the ODE

$$\mathrm{d}Z_t = v(Z_t, t)\,\mathrm{d}t, \qquad Z_0 \sim \pi_0,$$

whose field is fitted along the straight line between the endpoints,

$$\min_v \int_0^1 \mathbb{E}\Bigl[ \bigl\| (X_1 - X_0) - v(X_t, t) \bigr\|^2 \Bigr]\,\mathrm{d}t, \qquad X_t = t X_1 + (1 - t) X_0 .$$

This is the paper's eq. (1), and its minimiser is the conditional expectation $v^X(x, t) = \mathbb{E}[X_1 - X_0 \mid X_t = x]$.

Relation to what came before. The objective is exactly Flow Matching's optimal-transport path at $\sigma_{\min} = 0$: the interpolation is the same, and the regression target $X_1 - X_0$ is the same and equally independent of $t$. What Rectified Flow adds is not a different loss but what is done with the trained model.

Marginal preservation (Theorem 3.3). The solve is not merely a sampler that happens to work — $\mathrm{Law}(Z_t) = \mathrm{Law}(X_t)$ at every $t$, so $Z_1$ is distributed as the data while $(Z_0, Z_1)$ is a different, non-independent coupling of the same two marginals. That is what makes the next step legitimate.

Reflow. Because the endpoints of the solve are themselves a valid coupling, the whole procedure can be applied to its own output:

$$\mathbf{Z}^{k+1} = \mathtt{RectFlow}\bigl((Z_0^k, Z_1^k)\bigr), \qquad (Z_0^0, Z_1^0) = (X_0, X_1).$$

Each round trains on the previous model's (noise, sample) pairs instead of independent draws. The marginals never move — every $\mathbf{Z}^k$ generates the same data distribution — but the transport cost never increases and the paths get straighter.

Straightness is what improves, and it is measurable:

$$S(\mathbf{Z}) = \int_0^1 \mathbb{E}\Bigl[ \bigl\| (Z_1 - Z_0) - \dot{Z}_t \bigr\|^2 \Bigr]\,\mathrm{d}t ,$$

zero exactly when every trajectory is a straight line traversed at constant speed. A straight flow is one the forward Euler method integrates without discretisation error, so $S \to 0$ is precisely the statement that one step suffices — which is why the paper reports a state-of-the-art one-step FID of 4.85 on CIFAR-10.

Distillation is the same loss with $t$ pinned rather than drawn: regressing $v(\cdot, 0)$ onto $Z_1 - Z_0$ over the reflow pairs makes $z_0 + v(z_0, 0)$ a direct one-step map. Reflow is what makes that regression easy — it is fitting an already nearly-straight path, not shortcutting a curved one.

Time runs from noise to data: $t = 0$ is $\pi_0$ and $t = 1$ is $\pi_1$.