# On Investigating the Conservative Property of Score-Based Generative Models

arXiv: https://arxiv.org/abs/2209.12753 · HTML: https://mlsanity.com/paper/2209.12753/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 24 of its 48 references that are in the map.
- First posted on arXiv 26 Sep 2022.

## Placed in

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

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [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)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [A Note on the Inception Score](https://mlsanity.com/paper/1801.01973/) (2018)
- [Poisson Flow Generative Models](https://mlsanity.com/paper/2209.11178/) (2022)
- [BDDM: Bilateral Denoising Diffusion Models for Fast and High-Quality Speech Synthesis](https://mlsanity.com/paper/2203.13508/) (2022)
- [Denoising Likelihood Score Matching for Conditional Score-based Data Generation](https://mlsanity.com/paper/2203.14206/) (2022)
- [Deep Energy Estimator Networks](https://mlsanity.com/paper/1805.08306/) (2018)
- [On approximating nabla f with neural networks](https://mlsanity.com/paper/1910.12744/) (2019)
- [Conservativeness of untied auto-encoders](https://mlsanity.com/paper/1506.07643/) (2015)

## Built on by (most cited first, 16 of 16)

- [Consistent Diffusion Models: Mitigating Sampling Drift by Learning to be Consistent](https://mlsanity.com/paper/2302.09057/) (2023)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Characteristic Guidance: Non-linear Correction for Diffusion Model at Large Guidance Scale](https://mlsanity.com/paper/2312.07586/) (2023)
- [On gauge freedom, conservativity and intrinsic dimensionality estimation in diffusion models](https://mlsanity.com/paper/2402.03845/) (2024)
- [A Survey on Diffusion Models for Recommender Systems](https://mlsanity.com/paper/2409.05033/) (2024)
- [Beyond Masked and Unmasked: Discrete Diffusion Models via Partial Masking](https://mlsanity.com/paper/2505.18495/) (2025)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [Probability Density Geodesics in Image Diffusion Latent Space](https://mlsanity.com/paper/2504.06675/) (2025)
- [When Diffusion Models Memorize: Inductive Biases in Probability Flow of Minimum-Norm Shallow Neural Nets](https://mlsanity.com/paper/2506.19031/) (2025)
- [DiffAug: A Diffuse-and-Denoise Augmentation for Training Robust Classifiers](https://mlsanity.com/paper/2306.09192/) (2023)
- [Minimal Impact ControlNet: Advancing Multi-ControlNet Integration](https://mlsanity.com/paper/2506.01672/) (2025)
- [GeodesicNVS: Probability Density Geodesic Flow Matching for Novel View Synthesis](https://mlsanity.com/paper/2603.01010/) (2026)
- [Diffusion Models Observe Only Gradients: A Geometric Perspective on Score Matching Errors](https://mlsanity.com/paper/2606.06179/) (2026)
- [Classification-Denoising Networks](https://mlsanity.com/paper/2410.03505/) (2024)
- [Matrix-Free Least Squares Solvers: Values, Gradients, and What to Do With Them](https://mlsanity.com/paper/2510.19634/) (2025)
- [Conservative Flows: A New Paradigm of Generative Models](https://mlsanity.com/paper/2605.06905/) (2026)

## Abstract (arXiv)

Existing Score-Based Models (SBMs) can be categorized into constrained SBMs (CSBMs) or unconstrained SBMs (USBMs) according to their parameterization approaches. CSBMs model probability density functions as Boltzmann distributions, and assign their predictions as the negative gradients of some scalar-valued energy functions. On the other hand, USBMs employ flexible architectures capable of directly estimating scores without the need to explicitly model energy functions. In this paper, we demonstrate that the architectural constraints of CSBMs may limit their modeling ability. In addition, we show that USBMs' inability to preserve the property of conservativeness may lead to degraded performance in practice. To address the above issues, we propose Quasi-Conservative Score-Based Models (QCSBMs) for keeping the advantages of both CSBMs and USBMs. Our theoretical derivations demonstrate that the training objective of QCSBMs can be efficiently integrated into the training processes by leveraging the Hutchinson's trace estimator. In addition, our experimental results on the CIFAR-10, CIFAR-100, ImageNet, and SVHN datasets validate the effectiveness of QCSBMs. Finally, we justify the advantage of QCSBMs using an example of a one-layered autoencoder.