Google Research Finally Explains Why Diffusion Models Don't Just Copy Training Data

Diffusion models are among the most powerful tools for generative tasks requiring complex structure, like image generation and molecular discovery, and they have shown an exciting ability to generalize beyond their training data. But nobody could fully explain why. A new paper from Google Research, presented at ICLR 2026, now offers a rigorous mathematical answer.

The core finding is that a model's creativity is not a random fluke. Instead, it is a consequence of how neural network training naturally "smooths" the transformation from noise back to data during the generation process. In short: the very imperfection of how neural networks learn is what makes diffusion models creative.

The memorization trap

Training a diffusion model begins with taking real data samples and intentionally corrupting them with noise until they become unrecognizable. The model is then trained to reverse this corruption step-by-step, a process called denoising. If the model learns to do this perfectly based only on its training samples, it should produce carbon copies of them during deployment , a behavior known as memorization.

The scientific community has been divided on whether these models are truly creative or merely imitate what they have seen. An even more pressing concern is memorization, where the model's output partially or fully resembles training data, raising serious implications for data privacy. The memorization phenomenon has been observed in practice when models have large capacities relative to the training set size, which likely results in too good an approximation to the empirical score function.

So what stops a perfectly-trained model from just being a very expensive copy machine? The answer turns out to live in the mathematics of the score function , the learned "force field" that guides noisy data back toward meaningful outputs during generation.

Score smoothing: the accidental engine of creativity

The score function (SF) is the gradient of the log-probability of the data at any given noise level. Think of it as a vector field where every point in noisy space has an arrow pointing toward the nearest plausible image. At the core of diffusion models is the training of neural networks to fit a series of target functions called the empirical score functions (ESFs), which drive the denoising process at inference time. When equipped with the exact ESF instead of the version learned by neural networks, the diffusion model ends up generating data points that already exist in the training set , memorization.