# Learning ω-Powered Conditional Distributions with Importance-Weighted Flow Matching

arXiv: https://arxiv.org/abs/2610.16419 · HTML: https://mlsanity.com/paper/2610.16419/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 34 of its 42 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 34)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
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- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Exploring CLIP for Assessing the Look and Feel of Images](https://mlsanity.com/paper/2207.12396/) (2022)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022)
- [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020)
- [Guiding a Diffusion Model with a Bad Version of Itself](https://mlsanity.com/paper/2406.02507/) (2024)
- [Blind Image Quality Assessment via Vision-Language Correspondence: A Multitask Learning Perspective](https://mlsanity.com/paper/2303.14968/) (2023)
- [Exposing flaws of generative model evaluation metrics and their unfair treatment of diffusion models](https://mlsanity.com/paper/2306.04675/) (2023)
- [Flow Matching in Latent Space](https://mlsanity.com/paper/2307.08698/) (2023)
- [CFG++: Manifold-constrained Classifier Free Guidance for Diffusion Models](https://mlsanity.com/paper/2406.08070/) (2024)
- [On Memorization in Diffusion Models](https://mlsanity.com/paper/2310.02664/) (2023)
- [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Classifier-Free Guidance is a Predictor-Corrector](https://mlsanity.com/paper/2408.09000/) (2024)
- [On Nesting Monte Carlo Estimators](https://mlsanity.com/paper/1709.06181/) (2017)
- [Energy-Weighted Flow Matching for Offline Reinforcement Learning](https://mlsanity.com/paper/2503.04975/) (2025)
- [What does guidance do? A fine-grained analysis in a simple setting](https://mlsanity.com/paper/2409.13074/) (2024)
- [Debiasing Guidance for Discrete Diffusion with Sequential Monte Carlo](https://mlsanity.com/paper/2502.06079/) (2025)

## Abstract (arXiv)

Classifier-Free Guidance (CFG) with guidance scale γ is commonly motivated by the ω-powered conditional distribution obtained by setting ω=γ, but CFG does not, in general, sample from this distribution exactly. We study whether the powered target can instead be learned explicitly. We introduce Importance-Weighted Conditional Flow Matching (IW-CFM), which reweights conditional flow-matching examples using density ratios obtained from frozen conditional and unconditional flow models. With exact normalized weights, the resulting objective has the same gradient as flow matching on the powered target; in practice, we use self-normalized estimation and a regularized empirical approximation for high-dimensional data. On synthetic distributions, IW-CFM recovers analytical powered targets for which numerically matched CFG, γ=ω, exhibits a visible mismatch. On ImageNet, an amortized lightweight corrector achieves better alignment than fixed- and matched-guidance CFG with the corresponding weighted and filtered reference distributions. We additionally present an exploratory CLIP analysis of powered-density rankings on real ImageNet data. Aggregate shifts across selected prompt-specific contrasts coexist with heterogeneous classwise responses, including nonmonotonic trajectories. These results provide a practical route for studying and approximating model-defined powered conditional distributions.