# U-Turn Diffusion

arXiv: https://arxiv.org/abs/2308.07421 · HTML: https://mlsanity.com/paper/2308.07421/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 14 Aug 2023.

## Placed in

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

## Builds on (most cited first, 16 of 16)

- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Generative Modeling with Denoising Auto-Encoders and Langevin Sampling](https://mlsanity.com/paper/2002.00107/) (2020)
- [Spontaneous Symmetry Breaking in Generative Diffusion Models](https://mlsanity.com/paper/2305.19693/) (2023)
- [Boomerang: Local sampling on image manifolds using diffusion models](https://mlsanity.com/paper/2210.12100/) (2022)
- [Quantized Compressed Sensing with Score-based Generative Models](https://mlsanity.com/paper/2211.13006/) (2022)

## Built on by (most cited first, 8 of 8)

- [Probing the Latent Hierarchical Structure of Data via Diffusion Models](https://mlsanity.com/paper/2410.13770/) (2024)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [Theory of Speciation Transitions in Diffusion Models with General Class Structure](https://mlsanity.com/paper/2602.04404/) (2026)
- [Optimizing Noise Schedules of Generative Models in High Dimensionss](https://mlsanity.com/paper/2501.00988/) (2025)
- [Sampling Data with Chains of Forward-Backward Diffusion Steps](https://mlsanity.com/paper/2605.27006/) (2026)
- [Adaptive Path Integral Diffusion: AdaPID](https://mlsanity.com/paper/2512.11858/) (2025)
- [Analytic Bridge Diffusions for Controlled Path Generation](https://mlsanity.com/paper/2605.02961/) (2026)
- [Weighting Schedules Govern What and When Score-Based Generative Models Learn from Multimodal Data](https://mlsanity.com/paper/2609.35322/) (2026)

## Abstract (arXiv)

We investigate diffusion models generating synthetic samples from the probability distribution represented by the Ground Truth (GT) samples. We focus on how GT sample information is encoded in the Score Function (SF), computed (not simulated) from the Wiener-Ito (WI) linear forward process in the artifical time t∈ [0→ ∞], and then used as a nonlinear drift in the simulated WI reverse process with t∈ [∞→ 0]. We propose U-Turn diffusion, an augmentation of a pre-trained diffusion model, which shortens the forward and reverse processes to t∈ [0→ Tu] and t∈ [Tu→ 0]. The U-Turn reverse process is initialized at Tu with a sample from the probability distribution of the forward process (initialized at t=0 with a GT sample) ensuring a detailed balance relation between the shorten forward and reverse processes. Our experiments on the class-conditioned SF of the ImageNet dataset and the multi-class, single SF of the CIFAR-10 dataset reveal a critical Memorization Time  Tm , beyond which generated samples diverge from the GT sample used to initialize the U-Turn scheme, and a Speciation Time  Ts , where for  Tu > Ts > Tm , samples begin representing different classes. We further examine the role of SF non-linearity through a Gaussian Test, comparing empirical and Gaussian-approximated U-Turn auto-correlation functions, and showing that the SF becomes effectively affine for  t > Ts , and approximately affine for t∈ [Tm,Ts].