Towards demystifying the creativity of diffusion models
- source_url: https://research.google/blog/towards-demystifying-the-creativity-of-diffusion-models/
- published: 2026-07-15
- author: Zhengdao Chen, Research Scientist, Google Research
- source: Google Research Blog
- labels: Algorithms & Theory; Generative AI; Machine Intelligence
Captured article text
We show that a diffusion model’s creativity (its ability to generate novel data, rather than just memorize its training set) is a mathematical consequence of neural networks learning a “smoothed” version of the score function, driving the model to interpolate between training data points along the hidden data manifold.
Diffusion models are currently one of the most powerful types of tools for generative tasks that require complex and local structures, such as image generation and molecular discovery. They have shown an ability to generalize beyond their training data and, in this sense, exhibit “creativity”. For instance, after being trained with datasets of actual images, they can transform random noise samples into novel, high-quality images.
While this creative capability is impressive, it raises an important question: where does it come from? Understanding the answer is an important step towards demystifying the black-box nature of diffusion-based generative AI.
In “On the Interpolation Effect of Score Smoothing in Diffusion Models”, presented at ICLR 2026, the authors study the mathematics of diffusion models. They show that a model’s creativity is a consequence of how neural network training naturally “smooths” the transformation from noise back to the data during generation.
Understanding denoising
Training a diffusion model begins with taking real training data samples, such as cat photos, and intentionally corrupting them with noise until they become unrecognizable. The model is trained to reverse this corruption step by step so that it can reconstruct a realistic-looking image from pure noise. This is called denoising.
If the model learned to perform this denoising process perfectly from the training samples, it should produce copies of them at deployment time, a behavior known as memorization. In that scenario, the model would act as a retrieval tool rather than a creative engine capable of generating novel outputs. In practice, diffusion models usually do more than memorize; they generalize to generate new data samples.
To understand how diffusion models denoise data, imagine random noise as a cloud of gas particles scattered across a room, where a force field pulls each particle in a specific direction until the particles form a meaningful shape. In a diffusion model, the moving particles are data points undergoing denoising. The force field is the score function (SF), which is learned from the training data and dictates where the particles should flow at any moment.
If the model relied on a score function learned perfectly from the training data, the force field would drive particles into positions that exactly replicate the training data points, resulting in memorization.
Diffusion model creativity: The 1-dimension example
The research finds that diffusion model creativity originates from the approximate way neural networks learn. Imperfect training due to regularization naturally leads to a slight blurring of the learned score function, a process called “score smoothing”. This causes the denoising process to generate data that interpolates, or falls in the space between, the training points, creating new and plausible data samples.
Imagine a one-dimensional world with only two training data points: +1 and -1. At late stages of denoising, the perfect score function has a steep change of sign halfway between the two points. Particles on the left are pulled towards -1 and particles on the right towards +1. Eventually every particle converges to one of the training data points, so memorization occurs.
In practice, diffusion models do not have access to the perfect score function but use an approximate version learned by a neural network. Because of the regularization effect of weight decay, neural networks have difficulty learning functions with sharp cliffs. They instead learn smoother versions of the perfect score function, softening the steep drop into a gentler slope.
The authors train two-layer ReLU neural networks to fit the score function in a one-dimensional example, optimizing the parameters with AdamW under varying degrees of weight decay. The stronger the weight decay, the smoother the learned score function is in the middle area. Particles in that region flow more slowly and eventually rest within the interpolation zone between the two training data points.
The paper quantifies this connection by combining function-space theory of neural network regularization with the mathematics of denoising. Experiments also show that score smoothing can result from implicit regularization in neural networks trained by gradient-based algorithms, even without explicit strategies such as weight decay.
Score smoothing facilitates manifold recovery
In the real world, complex data such as high-resolution images live in high-dimensional pixel spaces. Most of that space is random noise that is meaningless to people. Only a small fraction corresponds to recognizable images, and those points live on a data manifold. The shape and location of this manifold are not known in advance, so image generation can be viewed as manifold recovery: the model infers the hidden manifold from finite training samples and generates new points on it that correspond to meaningful images.
Score smoothing is crucial for this process. In multi-dimensional settings, its effect is direction-dependent. Along directions parallel, or tangential, to the hidden data manifold, it produces a slowing-down effect similar to the one-dimensional case. Along directions pointing towards the manifold, the perfect score function is already relatively smooth, and further smoothing makes little difference.
Therefore, score smoothing does not slow particle movement toward the manifold, which would make final images blurry. It mainly reduces the tendency to collapse onto training data along tangential directions. The model can thus balance quality and novelty: outputs reach the meaningful data manifold while settling into blank spaces between the original training samples.
Conclusion
The findings suggest that what we call the creativity of diffusion models might be a predictable mathematical result. Because neural networks are never perfectly sharp, they create bridges that interpolate between known data. In image generation or drug discovery, a diffusion model may not simply remember two cat images or drug molecules; it can explore the space around them to suggest a third, new image or molecular configuration combining traces of both.
This work is an initial effort, and it remains to be seen what happens when data distributions or neural network architectures become more complex. The result suggests that models could be intentionally built as better interpolators, preserving creative generation while avoiding blind memorization. The authors also released code for the numerical experiments used to generate the paper’s figures.