# Trajectory Consistency Distillation: Improved Latent Consistency Distillation by Semi-Linear Consistency Function with Trajectory Mapping

arXiv: https://arxiv.org/abs/2402.19159 · HTML: https://mlsanity.com/paper/2402.19159/

## Key facts

- Cited by 46 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 44 of its 56 references that are in the map.
- First posted on arXiv 29 Feb 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 44)

- [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)
- [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)
- [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)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [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)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [AnimateDiff: Animate Your Personalized Text-to-Image Diffusion Models without Specific Tuning](https://mlsanity.com/paper/2307.04725/) (2023)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [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)
- [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)
- [eDiff-I: Text-to-Image Diffusion Models with an Ensemble of Expert Denoisers](https://mlsanity.com/paper/2211.01324/) (2022)
- [Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference](https://mlsanity.com/paper/2310.04378/) (2023)
- [Adversarial Diffusion Distillation](https://mlsanity.com/paper/2311.17042/) (2023)
- [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)

## Built on by (most cited first, 30 of 46)

- [Improved Distribution Matching Distillation for Fast Image Synthesis](https://mlsanity.com/paper/2405.14867/) (2024)
- [Fast High-Resolution Image Synthesis with Latent Adversarial Diffusion Distillation](https://mlsanity.com/paper/2403.12015/) (2024)
- [Hyper-SD: Trajectory Segmented Consistency Model for Efficient Image Synthesis](https://mlsanity.com/paper/2404.13686/) (2024)
- [Phased Consistency Models](https://mlsanity.com/paper/2405.18407/) (2024)
- [One-Step Diffusion Distillation through Score Implicit Matching](https://mlsanity.com/paper/2410.16794/) (2024)
- [A Survey on Diffusion Models for Time Series and Spatio-Temporal Data](https://mlsanity.com/paper/2404.18886/) (2024)
- [EM Distillation for One-step Diffusion Models](https://mlsanity.com/paper/2405.16852/) (2024)
- [Adversarial Diffusion Compression for Real-World Image Super-Resolution](https://mlsanity.com/paper/2411.13383/) (2024)
- [SF-V: Single Forward Video Generation Model](https://mlsanity.com/paper/2406.04324/) (2024)
- [OSV: One Step is Enough for High-Quality Image to Video Generation](https://mlsanity.com/paper/2409.11367/) (2024)
- [Autoregressive Diffusion Transformer for Text-to-Speech Synthesis](https://mlsanity.com/paper/2406.05551/) (2024)
- [Diff-Instruct++: Training One-step Text-to-image Generator Model to Align with Human Preferences](https://mlsanity.com/paper/2410.18881/) (2024)
- [DITTO-2: Distilled Diffusion Inference-Time T-Optimization for Music Generation](https://mlsanity.com/paper/2405.20289/) (2024)
- [Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective](https://mlsanity.com/paper/2405.16418/) (2024)
- [Dual-Expert Consistency Model for Efficient and High-Quality Video Generation](https://mlsanity.com/paper/2506.03123/) (2025)
- [Connecting Consistency Distillation to Score Distillation for Text-to-3D Generation](https://mlsanity.com/paper/2407.13584/) (2024)
- [NitroFusion: High-Fidelity Single-Step Diffusion through Dynamic Adversarial Training](https://mlsanity.com/paper/2412.02030/) (2024)
- [Flash-DMD: Towards High-Fidelity Few-Step Image Generation with Efficient Distillation and Joint Reinforcement Learning](https://mlsanity.com/paper/2511.20549/) (2025)
- [Instant Adversarial Purification with Adversarial Consistency Distillation](https://mlsanity.com/paper/2408.17064/) (2024)
- [Improved Training Technique for Latent Consistency Models](https://mlsanity.com/paper/2502.01441/) (2025)
- [Normalized Attention Guidance: Universal Negative Guidance for Diffusion Models](https://mlsanity.com/paper/2505.21179/) (2025)
- [RayFlow: Instance-Aware Diffusion Acceleration via Adaptive Flow Trajectories](https://mlsanity.com/paper/2503.07699/) (2025)
- [UniLumos: Fast and Unified Image and Video Relighting with Physics-Plausible Feedback](https://mlsanity.com/paper/2511.01678/) (2025)
- [David and Goliath: Small One-step Model Beats Large Diffusion with Score Post-training](https://mlsanity.com/paper/2410.20898/) (2024)
- [AutoLoRA: AutoGuidance Meets Low-Rank Adaptation for Diffusion Models](https://mlsanity.com/paper/2410.03941/) (2024)
- [PLADIS: Pushing the Limits of Attention in Diffusion Models at Inference Time by Leveraging Sparsity](https://mlsanity.com/paper/2503.07677/) (2025)
- [LiveLight: Real-time Streaming Video Relighting with Interactive Control](https://mlsanity.com/paper/2608.01771/) (2026)
- [Differentiable Solver Search for Fast Diffusion Sampling](https://mlsanity.com/paper/2505.21114/) (2025)
- [One-Way Ticket:Time-Independent Unified Encoder for Distilling Text-to-Image Diffusion Models](https://mlsanity.com/paper/2505.21960/) (2025)
- [FADA: Fast Diffusion Avatar Synthesis with Mixed-Supervised Multi-CFG Distillation](https://mlsanity.com/paper/2412.16915/) (2024)

## Abstract (arXiv)

Latent Consistency Model (LCM) extends the Consistency Model to the latent space and leverages the guided consistency distillation technique to achieve impressive performance in accelerating text-to-image synthesis. However, we observed that LCM struggles to generate images with both clarity and detailed intricacy. Consequently, we introduce Trajectory Consistency Distillation (TCD), which encompasses trajectory consistency function and strategic stochastic sampling. The trajectory consistency function diminishes the parameterisation and distillation errors by broadening the scope of the self-consistency boundary condition with trajectory mapping and endowing the TCD with the ability to accurately trace the entire trajectory of the Probability Flow ODE in semi-linear form with an Exponential Integrator. Additionally, strategic stochastic sampling provides explicit control of stochastic and circumvents the accumulated errors inherent in multi-step consistency sampling. Experiments demonstrate that TCD not only significantly enhances image quality at low NFEs but also yields more detailed results compared to the teacher model at high NFEs.