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This research investigates the theoretical mechanisms behind diffusion model creativity, proposing that neural networks learn a smoothed score function which causes generated samples to interpolate training data rather than purely extrapolate. Through analytical solutions and numerical experiments on one-dimensional subspaces and nonlinear manifolds, the study demonstrates how this smoothing effect naturally emerges during denoising dynamics without requiring explicit regularization. The findings offer deeper insight into the fundamental mathematical properties governing generative capabilities in modern diffusion architectures.
On the Interpolation Effect of Score Smoothing in Diffusion Models
arXiv:2502.19499v4 Announce Type: replace Abstract: Diffusion models have achieved remarkable progress in various domains with an intriguing ability to produce new data that do not exist in the training set. In this work, we study the hypothesis that such creativity arises from the neural network backbone learning a smoothed version of the empirical score function, which guides the denoising dynamics to generate data points that interpolate the training data. Focusing mainly on settings where the training set lies uniformly in a one-dimensional subspace, we elucidate the interplay between score smoothing and the denoising dynamics with analytical solutions and numerical experiments, demonstrating how smoothing the score function can cause the denoised data samples to interpolate the training set along the subspace. Moreover, we present theoretical and empirical evidence that learning score functions with neural networks - either with or without explicit regularization - can naturally achieve a similar effect, including when the data belong to simple nonlinear manifolds.
Abstract:Diffusion models have achieved remarkable progress in various domains with an intriguing ability to produce new data that do not exist in the training set. In this work, we study the hypothesis that such creativity arises from the neural network backbone learning a smoothed version of the empirical score function, which guides the denoising dynamics to generate data points that interpolate the training data. Focusing mainly on settings where the training set lies uniformly in a one-dimensional subspace, we elucidate the interplay between score smoothing and the denoising dynamics with analytical solutions and numerical experiments, demonstrating how smoothing the score function can cause the denoised data samples to interpolate the training set along the subspace. Moreover, we present theoretical and empirical evidence that learning score functions with neural networks - either with or without explicit regularization - can naturally achieve a similar effect, including when the data belong to simple nonlinear manifolds.
35 pages, 14 figures. Code available at: this https URL
Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
arXiv:2502.19499 [cs.LG]
(or arXiv:2502.19499v4 [cs.LG] for this version)
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