Coreset-Induced Conditional Velocity Flow Matching

• The paper introduces a method that replaces the standard isotropic Gaussian source with a coreset-induced Gaussian surrogate to better capture multimodal conditional velocity distributions.
• It employs an entropic Sinkhorn coreset to compress data and lift atoms into a Gaussian mixture, significantly reducing the transport gap in hierarchical flow matching.
• Empirical results on MNIST, CIFAR-10, and ImageNet-32 show marked improvements in FID scores and efficiency compared to traditional HRF2 methods.

Coreset-Induced Conditional Velocity Flow Matching (CCVFM) is a generative model that augments hierarchical rectified flow with a data-informed source distribution. Its central claim is that, in hierarchical flow matching, the inner flow need not transport isotropic Gaussian noise to a multimodal conditional velocity law from scratch. Instead, CCVFM compresses the target distribution into weighted atoms using an entropic Sinkhorn coreset, lifts those atoms to a Gaussian mixture surrogate, induces from that surrogate a closed-form conditional velocity law, and then trains a lightweight correction flow only for the remaining surrogate-to-target residual (Wang et al., 13 May 2026). In this sense, CCVFM is a hierarchical flow-matching method whose defining intervention is source design in velocity space.

1. Conceptual setting

CCVFM is formulated inside the standard flow-matching and rectified-flow setup. One samples

$X_0 \sim \rho_0,\qquad X_1 \sim \rho_1,$

typically with

$\rho_0=\mathcal N(0,I_d),$

and defines the linear interpolation

$X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$

The associated velocity is

$V=X_1-X_0.$

In ordinary flow matching, the learned velocity field $u_\theta(x,t)$ is trained under squared loss, and its optimal predictor is

$u^\star(x,t)=E[V\mid X_t=x].$

The paper emphasizes that this predictor reproduces the correct marginal evolution of $X_t$ but collapses the full conditional velocity law $\pi(v\mid x,t)$ to its mean. When several endpoint completions are compatible with the same intermediate state, that averaging blurs local transport geometry, which the paper presents as one reason few-step generation is difficult (Wang et al., 13 May 2026).

Hierarchical rectified flow, specifically HRF2, addresses that issue by modeling the full conditional law

$\pi(v\mid x_t,t),$

using an inner flow in velocity space. The key identity is

$$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$

The outer flow evolves in data space, while the inner flow generates $\rho_0=\mathcal N(0,I_d),$0. CCVFM keeps this hierarchical formulation, but changes the inner source.

2. Why CCVFM replaces the inner Gaussian source

In HRF2, the inner flow usually starts from

$\rho_0=\mathcal N(0,I_d),$1

and transports that source to the conditional target law $\rho_0=\mathcal N(0,I_d),$2. CCVFM argues that this is intrinsically mismatched when $\rho_0=\mathcal N(0,I_d),$3 is multimodal, because the target conditional velocity law can itself be highly multimodal. At the generation boundary $\rho_0=\mathcal N(0,I_d),$4, the conditional law simplifies to

$\rho_0=\mathcal N(0,I_d),$5

so it is a translated copy of the data distribution. If $\rho_0=\mathcal N(0,I_d),$6 has complex multimodal structure, then an isotropic Gaussian inner source is being asked to create that structure from scratch (Wang et al., 13 May 2026).

CCVFM replaces this uninformed source by a surrogate built directly from a coreset of the target. The method first approximates the target distribution by a weighted atomic coreset, then smooths it into a Gaussian mixture model (GMM), and finally uses the GMM-induced conditional velocity law as the inner source. The learned transport is therefore no longer

$\rho_0=\mathcal N(0,I_d),$7

but rather

$\rho_0=\mathcal N(0,I_d),$8

where $\rho_0=\mathcal N(0,I_d),$9 is the conditional velocity law induced by the surrogate GMM. The paper’s interpretation is that the correction flow then crosses a much smaller transport gap and focuses on residual refinement instead of discovering modes de novo.

3. Coreset construction and Gaussian-mixture lifting

The target data law is $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$0, with empirical measure

$X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$1

Stage I of CCVFM approximates this empirical distribution by a weighted atomic measure

$X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$2

where $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$3 are anchor locations and $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$4, $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$5, are atom weights (Wang et al., 13 May 2026).

The coreset is obtained by solving a KL-regularized optimal-transport objective with a soft assignment matrix $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$6: $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$7 subject to

$X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$8

With $X_t=(1-t)X_0+tX_1,\qquad t\in[0,1].$9 fixed, the optimal assignments satisfy

$V=X_1-X_0.$0

followed by row normalization. The alternating updates are

$V=X_1-X_0.$1

The paper states that this stage costs $V=X_1-X_0.$2 per outer iteration (Wang et al., 13 May 2026).

The weighted atoms are then lifted into a GMM surrogate

$V=X_1-X_0.$3

with

$V=X_1-X_0.$4

The covariance is fitted from weighted residuals by truncated SVD. This GMM smoothing is not an auxiliary convenience; it is what makes the next stage analytically tractable, because HRF2 already provides a closed-form conditional-velocity identity for Gaussian-mixture targets.

4. Closed-form conditional velocity law and correction flow

Replacing $V=X_1-X_0.$5 by $V=X_1-X_0.$6 in the conditional identity yields a surrogate conditional velocity law

$V=X_1-X_0.$7

which the paper shows is itself a GMM: $V=X_1-X_0.$8 Its component parameters are

$V=X_1-X_0.$9

$u_\theta(x,t)$0

$u_\theta(x,t)$1

At $u_\theta(x,t)$2, the expression simplifies to

$u_\theta(x,t)$3

The surrogate conditional source can therefore be sampled exactly without a learned neural sampler (Wang et al., 13 May 2026).

Stage III trains the only neural component of the method: a correction flow in velocity space. One samples

$u_\theta(x,t)$4

defines the true target velocity

$u_\theta(x,t)$5

and interpolates

$u_\theta(x,t)$6

The correction network $u_\theta(x,t)$7 is trained with

$u_\theta(x,t)$8

The formal loss is the same flow-matching regression used in HRF2, but the source is different: HRF2 uses $u_\theta(x,t)$9, whereas CCVFM uses the exact surrogate law $u^\star(x,t)=E[V\mid X_t=x].$0 (Wang et al., 13 May 2026).

For inference, the paper recommends the one-outer-step sampler $u^\star(x,t)=E[V\mid X_t=x].$1. Given a correction budget $u^\star(x,t)=E[V\mid X_t=x].$2, one draws $u^\star(x,t)=E[V\mid X_t=x].$3, samples a surrogate velocity from the coreset-induced GMM, applies $u^\star(x,t)=E[V\mid X_t=x].$4 correction steps,

$u^\star(x,t)=E[V\mid X_t=x].$5

and returns

$u^\star(x,t)=E[V\mid X_t=x].$6

Total NFE is counted as $u^\star(x,t)=E[V\mid X_t=x].$7, with the initial closed-form surrogate sampling step counted as a zero-parameter NFE for fair comparison.

5. Theoretical characterization

The central transport quantity is

$u^\star(x,t)=E[V\mid X_t=x].$8

At $u^\star(x,t)=E[V\mid X_t=x].$9, the paper defines

$X_t$0

Its main transport-task reduction theorem states that

$X_t$1

Under the explicit compression assumption

$X_t$2

with $X_t$3, this yields

$X_t$4

By contrast, the isotropic Gaussian-source baseline satisfies

$X_t$5

Thus the CCVFM correction problem can shrink with coreset size, whereas the Gaussian-source analogue retains a positive dimension-scale lower bound (Wang et al., 13 May 2026).

The paper also analyzes the Stage III target magnitude through the conditional second moment of $X_t$6. Let

$X_t$7

and write the surrogate law as

$X_t$8

Then, with direct surrogate sampling,

$X_t$9

where $\pi(v\mid x,t)$0 is the average within-component covariance scale, $\pi(v\mid x,t)$1 the between-component spread, and $\pi(v\mid x,t)$2 the mean mismatch. For an independent Gaussian source,

$\pi(v\mid x,t)$3

The source-dependent excess ratio is therefore

$\pi(v\mid x,t)$4

This formalizes the intended advantage of CCVFM: the residual target is small when the surrogate conditional law is already close to the true conditional velocity law in mean and covariance (Wang et al., 13 May 2026).

6. Empirical performance

CCVFM is evaluated on MNIST, CIFAR-10, ImageNet-32, and CelebA-HQ 256, with FID as the main metric. The paper reports $\pi(v\mid x,t)$5 on MNIST, CIFAR-10, and ImageNet-32, and $\pi(v\mid x,t)$6 on CelebA-HQ (Wang et al., 13 May 2026).

Dataset	Setting	Reported result
MNIST	$\pi(v\mid x,t)$7, NFE $\pi(v\mid x,t)$8	$\pi(v\mid x,t)$9
CIFAR-10	$\pi(v\mid x_t,t),$0, NFE $\pi(v\mid x_t,t),$1	$\pi(v\mid x_t,t),$2
ImageNet-32	NFE $\pi(v\mid x_t,t),$3	HRF2 $\pi(v\mid x_t,t),$4, CCVFM $\pi(v\mid x_t,t),$5
CelebA-HQ 256	CCVFM-L, $\pi(v\mid x_t,t),$6	$\pi(v\mid x_t,t),$7

On MNIST, the improvement over HRF2 is especially large. The table in the paper reports HRF2 at $\pi(v\mid x_t,t),$8 with $\pi(v\mid x_t,t),$9 NFE, while CCVFM reaches $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$0 at $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$1 NFE and $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$2 at $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$3 NFE. On ImageNet-32, where the paper emphasizes matched dataset, architecture, and protocol, CCVFM improves over HRF2 from $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$4 to $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$5 at $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$6 NFE, from $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$7 to $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$8 at $$\pi(v\mid x_t,t)\propto \rho_0(x_t-tv)\,\rho_1(x_t+(1-t)v).$$9 NFE, and from $\rho_0=\mathcal N(0,I_d),$00 to $\rho_0=\mathcal N(0,I_d),$01 at $\rho_0=\mathcal N(0,I_d),$02 NFE (Wang et al., 13 May 2026).

The coreset-size ablations are monotone in the reported settings. On MNIST, increasing $\rho_0=\mathcal N(0,I_d),$03 from $\rho_0=\mathcal N(0,I_d),$04 to $\rho_0=\mathcal N(0,I_d),$05 to $\rho_0=\mathcal N(0,I_d),$06 improves $\rho_0=\mathcal N(0,I_d),$07 from $\rho_0=\mathcal N(0,I_d),$08 to $\rho_0=\mathcal N(0,I_d),$09 to $\rho_0=\mathcal N(0,I_d),$10 under the listed correction budgets. On CIFAR-10, increasing $\rho_0=\mathcal N(0,I_d),$11 from $\rho_0=\mathcal N(0,I_d),$12 to $\rho_0=\mathcal N(0,I_d),$13 improves $\rho_0=\mathcal N(0,I_d),$14 from $\rho_0=\mathcal N(0,I_d),$15 to $\rho_0=\mathcal N(0,I_d),$16 at $\rho_0=\mathcal N(0,I_d),$17. The paper interprets these trends as qualitatively consistent with the Stage I compression assumption (Wang et al., 13 May 2026).

Ablations over correction depth and outer steps also support the method’s design. Larger $\rho_0=\mathcal N(0,I_d),$18 generally improves FID, with $\rho_0=\mathcal N(0,I_d),$19 best among the tested settings. By contrast, extra outer steps $\rho_0=\mathcal N(0,I_d),$20 do not help in the reported experiments; the paper recommends $\rho_0=\mathcal N(0,I_d),$21. Stage III contributes strongly on realistic image tasks, especially CelebA-HQ, where Stage II-only samples preserve coarse structure but remain blurrier than the corrected outputs. The paper also notes that on very simple 2D toy datasets, a one-step Stage III correction can be neutral or slightly harmful when Stage II already nearly saturates performance (Wang et al., 13 May 2026).

7. Position within the literature and stated limitations

CCVFM is positioned as a modification of hierarchical flow matching rather than as a replacement for the outer flow formalism. Relative to standard flow matching and rectified flow, its distinction is that it does not settle for the conditional mean velocity $\rho_0=\mathcal N(0,I_d),$22; it keeps the hierarchical objective of modeling the full conditional velocity law. Relative to HRF2, its novelty is narrower and more precise: it preserves the conditional-velocity identity but replaces the inner Gaussian source by a closed-form coreset-induced surrogate (Wang et al., 13 May 2026).

The method also sits adjacent to two broader lines of work. First, conditional Wasserstein formulations make condition-preserving transport explicit by restricting motion in conditioning coordinates (Chemseddine et al., 2024). Second, Conditional Variable Flow Matching learns flows over conditional density families by coupling state and conditioning variables jointly across a continuous conditioning manifold (Generale et al., 2024). CCVFM is distinct from both: it does not introduce a new conditional OT geometry over external conditioning variables, but instead redesigns the source law inside hierarchical velocity-space flow matching. A plausible implication is that it can be read as a source-design answer to the same general question raised elsewhere in flow matching, namely how much of the transport burden should be handled analytically before learning begins.

The paper states several limitations. The $\rho_0=\mathcal N(0,I_d),$23 benchmark is a classical quantization rate, but in high dimensions its constants are pessimistic. The crucial compression statement

$\rho_0=\mathcal N(0,I_d),$24

is assumed for the returned surrogate rather than proved for the deployed entropic Sinkhorn plus GMM-lift pipeline. The favorable NFE scaling also depends on additional regularity assumptions on the correction ODE. Performance depends on coreset quality and modality coverage, and Stage I introduces nontrivial preprocessing overhead. Empirically, CIFAR-10 still trails EDM in the cited comparison. Finally, the strongest formal guarantees are concentrated at the $\rho_0=\mathcal N(0,I_d),$25, $\rho_0=\mathcal N(0,I_d),$26 operating point, even though the paper also reports the more general nested sampler (Wang et al., 13 May 2026).