# Bidirectional Consistency Models

arXiv: https://arxiv.org/abs/2403.18035 · HTML: https://mlsanity.com/paper/2403.18035/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 54 of its 70 references that are in the map.
- First posted on arXiv 26 Mar 2024.

## Placed in

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

## Builds on (most cited first, 30 of 54)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [MobileNetV2: Inverted Residuals and Linear Bottlenecks](https://mlsanity.com/paper/1801.04381/) (2018)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Prompt-to-Prompt Image Editing with Cross Attention Control](https://mlsanity.com/paper/2208.01626/) (2022)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Null-text Inversion for Editing Real Images using Guided Diffusion Models](https://mlsanity.com/paper/2211.09794/) (2022)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference](https://mlsanity.com/paper/2310.04378/) (2023)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)

## Built on by (most cited first, 18 of 18)

- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [BELM: Bidirectional Explicit Linear Multi-step Sampler for Exact Inversion in Diffusion Models](https://mlsanity.com/paper/2410.07273/) (2024)
- [Consistency Diffusion Bridge Models](https://mlsanity.com/paper/2410.22637/) (2024)
- [Invertible Consistency Distillation for Text-Guided Image Editing in Around 7 Steps](https://mlsanity.com/paper/2406.14539/) (2024)
- [Improving Probabilistic Diffusion Models With Optimal Diagonal Covariance Matching](https://mlsanity.com/paper/2406.10808/) (2024)
- [PostEdit: Posterior Sampling for Efficient Zero-Shot Image Editing](https://mlsanity.com/paper/2410.04844/) (2024)
- [DiffRatio: Training One-Step Diffusion Models Without Teacher Supervision](https://mlsanity.com/paper/2502.08005/) (2025)
- [How to Guide Your Flow: Few-Step Alignment via Flow Map Reward Guidance](https://mlsanity.com/paper/2604.27147/) (2026)
- [Single-Step Bidirectional Unpaired Image Translation Using Implicit Bridge Consistency Distillation](https://mlsanity.com/paper/2503.15056/) (2025)
- [Efficient and Unbiased Sampling of Boltzmann Distributions via Consistency Models](https://mlsanity.com/paper/2409.07323/) (2024)
- [DEFT-VTON: Efficient Virtual Try-On with Consistent Generalised H-Transform](https://mlsanity.com/paper/2509.13506/) (2025)
- [Elucidating the Preconditioning in Consistency Distillation](https://mlsanity.com/paper/2502.02922/) (2025)
- [Beyond and Free from Diffusion: Invertible Guided Consistency Training](https://mlsanity.com/paper/2502.05391/) (2025)
- [VarDiU: A Variational Diffusive Upper Bound for One-Step Diffusion Distillation](https://mlsanity.com/paper/2508.20646/) (2025)
- [BiFM: Bidirectional Flow Matching for Few-Step Image Editing and Generation](https://mlsanity.com/paper/2603.24942/) (2026)
- [Inconsistencies In Consistency Models: Better ODE Solving Does Not Imply Better Samples](https://mlsanity.com/paper/2411.08954/) (2024)

## Abstract (arXiv)

Diffusion models (DMs) are capable of generating remarkably high-quality samples by iteratively denoising a random vector, a process that corresponds to moving along the probability flow ordinary differential equation (PF ODE). Interestingly, DMs can also invert an input image to noise by moving backward along the PF ODE, a key operation for downstream tasks such as interpolation and image editing. However, the iterative nature of this process restricts its speed, hindering its broader application. Recently, Consistency Models (CMs) have emerged to address this challenge by approximating the integral of the PF ODE, largely reducing the number of iterations. Yet, the absence of an explicit ODE solver complicates the inversion process. To resolve this, we introduce Bidirectional Consistency Model (BCM), which learns a single neural network that enables both forward and backward traversal along the PF ODE, efficiently unifying generation and inversion tasks within one framework. We can train BCM from scratch or tune it using a pretrained consistency model, which reduces the training cost and increases scalability. We demonstrate that BCM enables one-step generation and inversion while also allowing the use of additional steps to enhance generation quality or reduce reconstruction error. We further showcase BCM's capability in downstream tasks, such as interpolation and inpainting. Our code and weights are available at https://github.com/Mosasaur5526/BCM-iCT-torch.