# Z2-Sampling: Zero-Cost Zigzag Trajectories for Semantic Alignment in Diffusion Models

arXiv: https://arxiv.org/abs/2604.23536 · HTML: https://mlsanity.com/paper/2604.23536/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Discrete and text diffusion, one of 14 directions in Diffusion models.
- Builds on 50 of its 53 references that are in the map.
- First posted on arXiv 26 Apr 2026.

## Placed in

- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)
- LLM alignment / RLHF / preference optimisation / [Diffusion and multimodal model alignment](https://mlsanity.com/alignment/diffusion-and-multimodal-model-alignment/)

## Builds on (most cited first, 30 of 50)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [SDXL: Improving Latent Diffusion Models for High-Resolution Image Synthesis](https://mlsanity.com/paper/2307.01952/) (2023)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Stable Video Diffusion: Scaling Latent Video Diffusion Models to Large Datasets](https://mlsanity.com/paper/2311.15127/) (2023)
- [CogVideoX: Text-to-Video Diffusion Models with An Expert Transformer](https://mlsanity.com/paper/2408.06072/) (2024)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [Variational Diffusion Models](https://mlsanity.com/paper/2107.00630/) (2021)
- [ImageReward: Learning and Evaluating Human Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2304.05977/) (2023)
- [Pick-a-Pic: An Open Dataset of User Preferences for Text-to-Image Generation](https://mlsanity.com/paper/2305.01569/) (2023)
- [GenEval: An Object-Focused Framework for Evaluating Text-to-Image Alignment](https://mlsanity.com/paper/2310.11513/) (2023)
- [Tune-A-Video: One-Shot Tuning of Image Diffusion Models for Text-to-Video Generation](https://mlsanity.com/paper/2212.11565/) (2022)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Human Preference Score v2: A Solid Benchmark for Evaluating Human Preferences of Text-to-Image Synthesis](https://mlsanity.com/paper/2306.09341/) (2023)
- [Diffusion Model Alignment Using Direct Preference Optimization](https://mlsanity.com/paper/2311.12908/) (2023)
- [ModelScope Text-to-Video Technical Report](https://mlsanity.com/paper/2308.06571/) (2023)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [Sora: A Review on Background, Technology, Limitations, and Opportunities of Large Vision Models](https://mlsanity.com/paper/2402.17177/) (2024)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Common Diffusion Noise Schedules and Sample Steps are Flawed](https://mlsanity.com/paper/2305.08891/) (2023)
- [Hunyuan-DiT: A Powerful Multi-Resolution Diffusion Transformer with Fine-Grained Chinese Understanding](https://mlsanity.com/paper/2405.08748/) (2024)

## Built on by (most cited first, 1 of 1)

- [GeoShrink: Accelerating Diffusion Transformers with Two Lines of Code](https://mlsanity.com/paper/2609.33723/) (2026)

## Abstract (arXiv)

Diffusion models have achieved unprecedented success in text-aligned generation, largely driven by Classifier-Free Guidance (CFG). However, standard CFG operates strictly on instantaneous gradients, omitting the intrinsic curvature of the data manifold. Recent methods like Zigzag-sampling (Z-Sampling) explicitly traverse multi-step forward-backward trajectories to probe this curvature, significantly improving semantic alignment. Yet, these explicit traversals triple the Neural Function Evaluation (NFE) cost and introduce unconstrained truncation errors from off-manifold evaluations, causing cumulative drift from the true marginal distribution. In this paper, we theoretically demonstrate that the explicit zigzag sequence is topologically reducible. We propose Implicit Z-Sampling, rigorously proving that intermediate states can be algebraically annihilated via operator dualities, physically eliminating off-manifold approximation errors. To push sampling efficiency to its theoretical lower bound, we introduce Z2-Sampling (Zero-cost Zigzag Sampling). Exploiting the Probability Flow ODE's temporal coherence, Z2-Sampling couples implicit algebraic collapse with a dynamically cached Temporal Semantic Surrogate. This restores the standard 2-NFE baseline without sacrificing semantic exploration. We formally prove via Backward Error Analysis that this discrete collapse inherently synthesizes a directional derivative curvature penalty. Finally, extensive evaluations demonstrate that Z2-Sampling structurally shatters the performance-efficiency Pareto frontier. We validate its universal applicability across diverse architectures (U-Nets, DiTs) and modalities (image/video), establishing seamless orthogonality with advanced alignment frameworks (AYS, Diffusion-DPO).