The conditional velocity is a posterior mean
Under linear interpolation $x_t = t x_1 + (1-t) x_0$, the state $x_t$ and the measurement $y = Ax_1 + \eta$
are two conditionally independent noisy observations of the same clean image $x_1$. Stacking them
gives an augmented observation model whose MMSE estimator is exactly the measurement-conditioned posterior
mean, and that mean determines the optimal conditional velocity:
$$ v(x_t, t, y) \;=\; \frac{\mathbb{E}[x_1 \mid x_t, y] - x_t}{1 - t}. $$
A variational characterization with explicit data consistency
Our main theoretical result (Proposition 1) shows that for a non-degenerate $p_1$ with bounded support,
$\sigma > 0$ and $t \in (0,1)$, there is a regularizer $\phi_t$—unique up to an additive constant—for which the posterior mean is the unique global minimizer and unique stationary point of
$$ \mathbb{E}[x_1 \mid x_t, y] \;=\; \arg\min_{x \in \mathbb{R}^n}\;
\frac{1}{2(1-t)^2}\lVert x_t - t x \rVert_2^2
\;+\; \frac{1}{2\sigma^2}\lVert y - A x \rVert_2^2
\;+\; \phi_t(x). $$
The first quadratic enforces interpolation consistency, the second enforces measurement consistency
explicitly, and $\phi_t$ absorbs whatever structure remains. We further prove that this velocity satisfies the
conditional continuity equation $\partial_t p_t(x \mid y) + \nabla \cdot \big(p_t(x \mid y) v(x,t,y)\big) = 0$,
so the induced flow transports the Gaussian source to $p_\tau(\cdot \mid y)$, which converges weakly to the
measurement-conditioned posterior $p(x_1 \mid y)$ as $\tau \to 1^-$. Data consistency becomes part of the
generative dynamics rather than an external sampling correction.
An operator-aware conditional velocity
$\phi_t$ is not available in closed form, so we split the objective with half-quadratic splitting and
alternate the two resulting updates. The $x$-update is a closed-form data-consistency solve that depends on
$A$ and $y$; the $z$-update combines interpolation consistency with the implicit regularizer and is replaced
by a learned estimator $R_\theta$, damped toward the data-consistent iterate. After $K$ iterations the output
$z_\theta^K(x_t, t, y)$ estimates the posterior mean and defines the learned velocity.
The whole estimator is trained end-to-end with a weighted flow-matching loss, using a stabilized time scale
$\tau_t = \max(1 - t, \tau_{\min})$ that keeps the velocity bounded near $t = 1$:
$$ \mathcal{L}(\theta) = \mathbb{E}_{x_1, x_0, t, y}\Big[\omega(t)\,
\lVert v_\theta(x_t,t,y) - v_{\text{target}} \rVert^2 \Big],
\qquad \omega(t) = \frac{1 + \tau_t^3}{\tau_t}. $$
The result is an operator-aware velocity field: every inner iteration of every velocity
evaluation applies $A$ and $A^{\top}$ through the data-consistency solve, so data consistency is learned as
part of the dynamics rather than imposed afterwards. Sampling therefore needs no guidance term, no
projection, and no proximal correction at inference—just $N$ explicit Euler steps.
Training
require: A, σ, [tmin, tmax], τmin
repeat
x₁ ~ p₁, x₀ ~ N(0,I), t ~ U[tmin, tmax]
η ~ N(0, σ²I), y ← A x₁ + η
xₜ ← t x₁ + (1−t) x₀, τₜ = max(1−t, τmin)
zK ← PostMean(xₜ, t, y, A)
vθ ← (zK − xₜ) / τₜ
vtarget ← (x₁ − xₜ) / τₜ
update θ with the weighted FM loss
until convergence
Sampling
require: y, A, trained model, step count N
x₀ ~ N(0, I)
Δt ← 1 / N
for i = 0 … N−1 do
tᵢ ← i Δt
zKi ← PostMean(xᵢ, tᵢ, y, A)
vᵢ ← (zKi − xᵢ) / (1 − tᵢ)
xᵢ⁺¹ ← xᵢ + Δt · vᵢ
return xN
no guidance, no projection, no proximal step
PostMean—operator-aware posterior mean
require: xₜ, t < 1, y, A, Rθ, K, ρ, β
z⁰ ← xₜ
for k = 0 … K−1 do
xᵏ⁺¹ ← (AᵀA + ρI)⁻¹ (Aᵀy + ρ zᵏ)
ẑᵏ⁺¹ ← Rθ(xᵏ⁺¹, xₜ, t)
zᵏ⁺¹ ← (1−β) xᵏ⁺¹ + β ẑᵏ⁺¹
return zK
θ enters only through Rθ; each call costs
K solves + K network evaluations
