arxiv: 2604.09181 · v1 · submitted 2026-04-10 · 💻 cs.CV · cs.LG

Recognition: unknown

MixFlow: Mixed Source Distributions Improve Rectified Flows

Nazir Nayal , Christopher Wewer , Jan Eric Lenssen

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:05 UTC · model grok-4.3

classification 💻 cs.CV cs.LG

keywords rectified flowsdiffusion modelsimage generationsource distributionsampling efficiencygenerative modelsflow matching

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The pith

Linear mixtures of unconditional and conditioned source distributions reduce curvature in rectified flows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that rectified flow models for image generation suffer from curved paths because their starting source distribution is independent of the data. It proposes training on linear mixtures of a standard Gaussian and a new conditioned source called kappa-FC to create better alignment from the start. This matters because straighter paths would let the same model reach high-quality outputs in fewer steps without added inference cost. The result is faster sampling and higher quality under a fixed budget.

Core claim

Rectified flow models trained with MixFlow optimize their velocity fields on linear combinations of an unconditional Gaussian source and a kappa-FC conditioned source. The mixture produces lower path curvature, tighter source-to-data alignment, quicker training convergence, and improved sample quality measured by FID.

What carries the argument

MixFlow training on linear mixtures of unconditional Gaussian and kappa-FC source distributions, which straightens the learned generative trajectories by improving initial alignment.

Load-bearing premise

The linear mixture reliably lowers path curvature without harming sample diversity or creating new training instabilities.

What would settle it

No measurable drop in FID or no reduction in required sampling steps when the same model is trained with MixFlow versus standard rectified flow on image benchmarks.

Figures

Figures reproduced from arXiv: 2604.09181 by Christopher Wewer, Jan Eric Lenssen, Nazir Nayal.

**Figure 1.** Figure 1: Method overview. We propose training rectified flows with mixed source distributions, obtained by interpolating a conditional and a simple unconditional distribution. The conditional distribution is predicted from a signal κ, which is possibly informative, e.g. a specific data example, a class label, or entirely independent, e.g. random noise. The learned conditional distribution provides a trajectory stru… view at source ↗

**Figure 2.** Figure 2: Qualitative Results. We show our method’s generation on FFHQ (rows 1-2) and AFHQv2 (rows 3-4) datasets using different steps. MixFlow requires few steps to generate reasonable outputs. 5.2 UNCONDITIONAL GENERATION ON FFHQ & AFHQ We further train and evaluate our method on higher resolution datasets FFHQ 64 × 64 Karras et al. (2019) and AFHQv2 64 × 64 Choi et al. (2020). We train the models with κ as the da… view at source ↗

**Figure 3.** Figure 3: Curvature vs. β. We show how the curvature of the generative trajectories changes with different β values. We can see a clear trend of lower curvature with a lower β value Parameters CIFAR10 FFHQ AFFHQv2 UNet Parameters Channel Size 128 128 128 Channel Multiplier [2,2,2] [1,2,2,2] [1,2,2,2] Blocks per Layer 4 4 4 Attention Resolution 16 16 16 Dropout Probability 0.13 0.05 0.25 Embedding type positional pos… view at source ↗

**Figure 4.** Figure 4: Effect of interpolation weight w. We visualize the effect of varying w (x-axis) during sampling on FID (y-axis) across different numbers of sampling steps (different lines). (a) Sampling with a few steps benefits from a larger weight of the source distribution conditioned on class label κc, while for many steps, the unconditional standard Gaussian is better suited. (b) With conditioning on uncorrelated Gau… view at source ↗

**Figure 5.** Figure 5: FID for Sampling Steps vs. weight w. We show the FID across different sampling step choices for both κc and κn as the interpolation parameter w changes. This is the numerical version of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: FID vs. training progress. Samples are generated with the RK45 solver across different step of the training process. Our method achieves the same performance as Fast-ODE (gray dotted line) with only 60% of the training budget. C.2 EFFECT OF w In the cases where the conditioning signals are available during inference, we can use them to vary the mixture parameter w while sampling. So to understand the impor… view at source ↗

**Figure 7.** Figure 7: Comparison against Rectified Flow. We show multiple examples for generated images with different sampling steps and compare against Rectified Flow. We highlight that for a low number of sampling steps (2,4), MixFlow generates much clearer images compared to Rectified Flow. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Qualitative Results on CIFAR10 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: Qualitative Results on FFHQ 64 × 64 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Qualitative Results on AFHQv2 64 × 64 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

read the original abstract

Diffusion models and their variations, such as rectified flows, generate diverse and high-quality images, but they are still hindered by slow iterative sampling caused by the highly curved generative paths they learn. An important cause of high curvature, as shown by previous work, is independence between the source distribution (standard Gaussian) and the data distribution. In this work, we tackle this limitation by two complementary contributions. First, we attempt to break away from the standard Gaussian assumption by introducing $\kappa\texttt{-FC}$, a general formulation that conditions the source distribution on an arbitrary signal $\kappa$ that aligns it better with the data distribution. Then, we present MixFlow, a simple but effective training strategy that reduces the generative path curvatures and considerably improves sampling efficiency. MixFlow trains a flow model on linear mixtures of a fixed unconditional distribution and a $\kappa\texttt{-FC}$-based distribution. This simple mixture improves the alignment between the source and data, provides better generation quality with less required sampling steps, and accelerates the training convergence considerably. On average, our training procedure improves the generation quality by 12\% in FID compared to standard rectified flow and 7\% compared to previous baselines under a fixed sampling budget. Code available at: $\href{https://github.com/NazirNayal8/MixFlow}{https://github.com/NazirNayal8/MixFlow}$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

MixFlow gets modest FID gains from mixing source distributions in rectified flows, but the claimed reduction in path curvature lacks direct measurements.

read the letter

The paper's core move is to replace the usual independent Gaussian source with a linear mix of that Gaussian and a κ-FC conditioned source that tries to align better with the data. They train the flow on these mixtures and report roughly 12% better FID than plain rectified flow and 7% better than prior baselines when the number of function evaluations is held fixed. That is the main empirical result and the part that could matter for people who care about sampling speed in flow models. The κ-FC formulation and the specific mixture training recipe look like the actual new pieces; earlier work had noted source-data independence as a problem, but this is a concrete way to act on it during training. The code release is also a plus for anyone who wants to check the numbers. The soft spot is exactly what the stress-test flagged: the authors attribute the gains to straighter paths, yet the abstract and the reported experiments do not include any direct curvature diagnostics such as integrated velocity norms, acceleration, or transport costs. Without those, or at least an ablation that isolates the mixture from the κ-FC term alone, it is hard to know whether the improvement comes from reduced curvature, from changed optimization dynamics, or from something else. Experimental details on baselines, seeds, and statistical significance are also thin in what is visible, so the 12% figure needs verification. This is the kind of incremental but practical tweak that people working on efficient generative sampling would want to see tested properly. It is not a foundational result, but the idea is simple enough that a careful referee could quickly tell whether the gains hold up and whether the mechanism story is supported. I would send it to review rather than desk-reject, mainly to get the missing ablations and metrics on the table.

Referee Report

2 major / 1 minor

Summary. The paper proposes κ-FC, a general formulation for conditioning the source distribution in rectified flows on an arbitrary signal κ to better align it with the data, and MixFlow, a training strategy that optimizes flow models on linear mixtures of a fixed unconditional source and a κ-FC source. It claims this mixture reduces generative path curvature, accelerates convergence, and yields average FID improvements of 12% over standard rectified flow and 7% over prior baselines under fixed sampling budgets, with code released.

Significance. If the reported FID gains are reproducible and the mechanism is confirmed, the work could offer a practical, low-overhead way to improve sampling efficiency in flow-based generative models without architectural changes. The public code release supports reproducibility and is a clear strength.

major comments (2)

[Experimental results] Experimental results section: the central claim attributes the 12% FID gain and faster convergence to reduced path curvature from the linear mixture, yet no direct curvature diagnostics (e.g., average ∫||v_t|| dt, integrated squared acceleration, or OT cost between effective source and data) are reported; FID at fixed NFE alone cannot isolate this mechanism from optimization or regularization effects.
[Ablation studies] Ablation studies: the manuscript lacks a controlled comparison of the full MixFlow mixture against training on the κ-FC component alone, so it is impossible to determine whether the reported gains require the mixture or are already achieved by the conditioning term.

minor comments (1)

[Abstract] Abstract: experimental details (dataset, model size, exact baselines, number of runs, statistical significance) are omitted, making it difficult to assess the strength of the 12% and 7% claims without the full text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive review and for recognizing the potential practical value of our approach along with the code release. We address each major comment below and will update the manuscript accordingly to strengthen the experimental support for our claims.

read point-by-point responses

Referee: Experimental results section: the central claim attributes the 12% FID gain and faster convergence to reduced path curvature from the linear mixture, yet no direct curvature diagnostics (e.g., average ∫||v_t|| dt, integrated squared acceleration, or OT cost between effective source and data) are reported; FID at fixed NFE alone cannot isolate this mechanism from optimization or regularization effects.

Authors: We agree that direct curvature diagnostics would provide stronger mechanistic evidence and help rule out confounding factors. The original manuscript relies on indirect indicators (FID at fixed NFEs, convergence speed, and qualitative path visualizations). In the revised version we will add quantitative metrics including average integrated velocity norm along trajectories and OT cost between the effective source and data distributions to better isolate the contribution of reduced curvature. revision: yes
Referee: Ablation studies: the manuscript lacks a controlled comparison of the full MixFlow mixture against training on the κ-FC component alone, so it is impossible to determine whether the reported gains require the mixture or are already achieved by the conditioning term.

Authors: We acknowledge that the current ablations do not include a direct head-to-head comparison of training on the κ-FC source in isolation versus the proposed linear mixture. While the paper already shows gains relative to standard rectified flow, adding this controlled ablation will clarify whether the mixture itself is necessary. We will include these results in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Empirical FID gains from mixture training; no derivation reduces to self-definition or fitted prediction

full rationale

The paper introduces κ-FC and MixFlow as a training procedure that mixes source distributions, then reports average 12% FID improvement over standard rectified flow under fixed sampling budgets. All central claims are supported by external experimental metrics on image datasets rather than any equation that defines the output in terms of itself or renames a fitted parameter as a prediction. The curvature-reduction mechanism is presented as a hypothesis supported by prior literature and empirical outcomes, with no self-citation chain or uniqueness theorem invoked to force the result. This is a standard non-circular empirical contribution.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are stated in the provided text. The mixing weight and choice of κ signal are likely hyperparameters but not detailed here.

pith-pipeline@v0.9.0 · 5546 in / 1007 out tokens · 53705 ms · 2026-05-10T17:05:19.940329+00:00 · methodology

discussion (0)

Reference graph

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