# High-Order Matching for One-Step Shortcut Diffusion Models

arXiv: https://arxiv.org/abs/2502.00688 · HTML: https://mlsanity.com/paper/2502.00688/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 56 of its 115 references that are in the map.
- First posted on arXiv 2 Feb 2025.

## 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 56)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Scaling Instruction-Finetuned Language Models](https://mlsanity.com/paper/2210.11416/) (2022)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [LLaMA-Adapter: Efficient Fine-tuning of Language Models with Zero-init Attention](https://mlsanity.com/paper/2303.16199/) (2023)
- [LLaMA-Adapter V2: Parameter-Efficient Visual Instruction Model](https://mlsanity.com/paper/2304.15010/) (2023)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [On the Importance of Noise Scheduling for Diffusion Models](https://mlsanity.com/paper/2301.10972/) (2023)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [An Over-parameterized Exponential Regression](https://mlsanity.com/paper/2303.16504/) (2023)
- [TokenCompose: Text-to-Image Diffusion with Token-level Supervision](https://mlsanity.com/paper/2312.03626/) (2023)
- [On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)](https://mlsanity.com/paper/2407.01079/) (2024)
- [Improved Noise Schedule for Diffusion Training](https://mlsanity.com/paper/2407.03297/) (2024)
- [Randomized and Deterministic Attention Sparsification Algorithms for Over-parameterized Feature Dimension](https://mlsanity.com/paper/2304.04397/) (2023)
- [LazyDiT: Lazy Learning for the Acceleration of Diffusion Transformers](https://mlsanity.com/paper/2412.12444/) (2024)
- [An Iterative Algorithm for Rescaled Hyperbolic Functions Regression](https://mlsanity.com/paper/2305.00660/) (2023)
- [How Far Are We From AGI: Are LLMs All We Need?](https://mlsanity.com/paper/2405.10313/) (2024)
- [A Fast Optimization View: Reformulating Single Layer Attention in LLM Based on Tensor and SVM Trick, and Solving It in Matrix Multiplication Time](https://mlsanity.com/paper/2309.07418/) (2023)
- [Flow matching achieves almost minimax optimal convergence](https://mlsanity.com/paper/2405.20879/) (2024)
- [Circuit Complexity Bounds for RoPE-based Transformer Architecture](https://mlsanity.com/paper/2411.07602/) (2024)
- [Numerical Pruning for Efficient Autoregressive Models](https://mlsanity.com/paper/2412.12441/) (2024)
- [Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective](https://mlsanity.com/paper/2405.16418/) (2024)
- [GradientCoin: A Peer-to-Peer Decentralized Large Language Models](https://mlsanity.com/paper/2308.10502/) (2023)

## Built on by (most cited first, 15 of 15)

- [T2VWorldBench: A Benchmark for Evaluating World Knowledge in Text-to-Video Generation](https://mlsanity.com/paper/2507.18107/) (2025)
- [Text-to-Image Diffusion Models Cannot Count, and Prompt Refinement Cannot Help](https://mlsanity.com/paper/2503.06884/) (2025)
- [On Computational Limits of FlowAR Models: Expressivity and Efficiency](https://mlsanity.com/paper/2502.16490/) (2025)
- [Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows](https://mlsanity.com/paper/2503.09069/) (2025)
- [Can You Count to Nine? A Human Evaluation Benchmark for Counting Limits in Modern Text-to-Video Models](https://mlsanity.com/paper/2504.04051/) (2025)
- [T2VTextBench: A Human Evaluation Benchmark for Textual Control in Video Generation Models](https://mlsanity.com/paper/2505.04946/) (2025)
- [Fast RoPE Attention: Combining the Polynomial Method and Fast Fourier Transform](https://mlsanity.com/paper/2505.11892/) (2025)
- [Only Large Weights (And Not Skip Connections) Can Prevent the Perils of Rank Collapse](https://mlsanity.com/paper/2505.16284/) (2025)
- [OAT-FM: Optimal Acceleration Transport for Improved Flow Matching](https://mlsanity.com/paper/2509.24936/) (2025)
- [Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria](https://mlsanity.com/paper/2508.07102/) (2025)
- [HOFAR: High-Order Augmentation of Flow Autoregressive Transformers](https://mlsanity.com/paper/2503.08032/) (2025)
- [Theoretical Foundation of Flow-Based Time Series Generation: Provable Approximation, Generalization, and Efficiency](https://mlsanity.com/paper/2503.14076/) (2025)
- [Imitation Learning Policy based on Multi-Step Consistent Integration Shortcut Model](https://mlsanity.com/paper/2510.19356/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)
- [VTV-FM: Flow Matching through Variational Terminal-Velocity Closure](https://mlsanity.com/paper/2610.00785/) (2026)

## Abstract (arXiv)

One-step shortcut diffusion models [Frans, Hafner, Levine and Abbeel, ICLR 2025] have shown potential in vision generation, but their reliance on first-order trajectory supervision is fundamentally limited. The Shortcut model's simplistic velocity-only approach fails to capture intrinsic manifold geometry, leading to erratic trajectories, poor geometric alignment, and instability-especially in high-curvature regions. These shortcomings stem from its inability to model mid-horizon dependencies or complex distributional features, leaving it ill-equipped for robust generative modeling. In this work, we introduce HOMO (High-Order Matching for One-Step Shortcut Diffusion), a game-changing framework that leverages high-order supervision to revolutionize distribution transportation. By incorporating acceleration, jerk, and beyond, HOMO not only fixes the flaws of the Shortcut model but also achieves unprecedented smoothness, stability, and geometric precision. Theoretically, we prove that HOMO's high-order supervision ensures superior approximation accuracy, outperforming first-order methods. Empirically, HOMO dominates in complex settings, particularly in high-curvature regions where the Shortcut model struggles. Our experiments show that HOMO delivers smoother trajectories and better distributional alignment, setting a new standard for one-step generative models.