July 15, 2026
Zhengdao Chen, Research Scientist, Google Research
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’ve shown an exciting capability 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 intriguing question: where does it come from? Understanding the answer to this question is an important step towards demystifying the “black-box” nature of diffusion-based generative AI. To that end, in "On the Interpolation Effect of Score Smoothing in Diffusion Models", presented at ICLR 2026, we dive into the mathematics of diffusion models to answer this question. We show that a model’s creativity isn’t a random fluke. Instead, it is a consequence of how neural network training naturally "smooths" the transformation from noise back to the data during the generation process.
Training a diffusion model begins with taking a set of real training data samples — like cat photos — and intentionally corrupting them with noise until they become completely unrecognizable. The model is then trained to reverse this corruption step-by-step so that it can reconstruct a realistic-looking image from pure noise, a process called denoising.
If the model learns to perform this denoising process perfectly based only on its training samples, it should produce carbon copies of them during deployment time as well (a behavior known as memorization). In this scenario, the model acts as a retrieval tool rather than as a creative engine capable of generating novel outputs. In practice, however, diffusion models usually do more than just memorize; they generalize to generate new data samples.
To understand how diffusion models actually 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 they form a meaningful shape. In a diffusion model, the moving particles are individual 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 given moment.
If the model relies on a score function learned perfectly from the training data, the force field will drive the particles into positions that exactly replicate the training data points (i.e., memorization). We discovered that the creativity of diffusion models actually originates from the approximate nature of how neural networks typically learn: imperfect training due to regularization naturally leads to a slight blurring of the learned score function in a process called “score smoothing”. This, in turn, causes the denoising process to generate data that interpolates (in other words, fall in the space between) the training points, thus creating new and plausible data samples.
Imagine a one-dimensional world with only two training data points: +1 and -1. At late stages of the denoising process, the “perfect” score function looks like the curvy gray line in the figure below, which has a steep change of sign halfway between the two points, meaning a rapid switch in the direction of pull near 0. In other words, the whole space is divided almost sharply into two, with particles on the left pulled towards -1 and particles on the right pulled towards +1. In the end, every particle converges to one of the two training data points, and thus, memorization occurs.
In practice, however, diffusion models don’t 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 during training, neural networks have a hard time learning functions with these sharp cliffs. Instead, they tend to learn smoother versions of the “perfect” score function, softening the steep drop into a gentler slope. To illustrate this, we set up an experiment to train two-layer ReLU neural networks to fit the score function in a 1-D example, with the neural networks’ parameters optimized by the popular AdamW algorithm under varying degrees of weight decay (WD).
The stronger the weight decay is, the smoother the learned score function is in the middle area, meaning that particles in that region flow slower than before and will eventually rest within the “interpolation zone” between the two training data points.
In the paper, we quantify this connection by combining the function-space theory of neural network regularization with the mathematics of denoising. Furthermore, our experiments also show that even without explicit regularization strategies such as weight decay, score smoothing can also result from the implicit regularization effect found in neural networks trained by gradient-based algorithms.
In the real world, complex data like high-resolution images live in high-dimensional pixel spaces rather than a simple 1-D world. The vast majority of that space, however, is just random noise that is meaningless to the human eye. Only a small fraction of the data points in that space correspond to recognizable images, and they live in what’s called the data manifold (like a sheet tucked inside a larger space). The shape and location of the data manifold are not known by the model in advance. Thus, image generation can be considered as a task of manifold recovery, where the model needs to infer what the hidden data manifold looks like based on the finite number of training data sampled from it, and then come up with new points on the manifold which will correspond to novel and meaningful images. It turns out that score smoothing is crucial for diffusion models to achieve this.
Remarkably, in multi-dimensional settings, the effect of score smoothing manifests in a direction-dependent manner. Along directions that are parallel (or “tangential”) to the hidden data manifold, it produces a similar slowing-down effect as in the 1-D scenario. However, along directions pointing towards the manifold, the "perfect" score function is already relatively smooth (in fact, just a straight line if the manifold is flat), and further smoothing does not make much difference.
Hence, instead of applying brakes to the particles' flow in every direction (which would stall them in the noisy empty space and result in the final images being blurry), score smoothing does not slow down their movement toward the manifold, but only reduces their tendency to collapse towards the training data along the tangential directions. In this way, the model achieves a balance between quality and novelty: the images are both realistic looking (because they successfully reached the meaningful data manifold) and new (because they settled into the blank spaces between the original training data points).
Our findings suggest that what we call the “creativity" of diffusion models might actually 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, this could mean the diffusion model doesn't just remember two different cat images or drug molecules it was shown; it explores the space around them to suggest a third, brand-new image or molecular configuration that combines traces of both.
Our work only serves as an initial effort towards elucidating this mechanism, and it remains to be seen what happens when the data distribution or the neural network architectures grow more complex. Nonetheless, by showing that this behavior is fundamentally rooted in how neural networks learn, we can begin to intentionally build models that are better "interpolators" to ensure they remain creative engines while avoiding the pitfalls of blind memorization. We have also released the code for the numerical experiments used to generate the figures in our paper.
We thank Sreenivas Gollapudi and Ravi Kumar for supporting the project as well as Mark Simborg and Kimberly Schwede for helping to prepare this blogpost.