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This paper investigates the generalization, memorization, and overfitting behaviors of diffusion models in the high-dimensional regime, specifically through the lens of denoising score matching in a vector-valued reproducing kernel Hilbert space. By deriving exact risk trajectories under gradient flow training, the authors identify three distinct phases of estimator behavior: a spectral estimator that generalizes, a pure-noise score that interpolates the training objective, and an empirical Bayes estimator that memorizes the data. This work not only draws parallels to supervised learning mechanisms but also uncovers unique phenomena pertinent to generative modeling, enhancing our understanding of how these models can effectively learn from high-dimensional data.
Generative models exhibit distinct phases of learning that reveal how they balance generalization and memorization in high-dimensional spaces.
Modern score-based generative models have achieved remarkable empirical success in high-dimensional tasks such as image, audio, and video synthesis. These models reduce distribution learning to a sequence of regression problems that, if solved exactly on finite data, would ultimately reproduce the training samples. Their ability to generalize must therefore arise from the implicit or explicit regularization during training. In this work, we develop a generative counterpart to the theory of benign overfitting and algorithmic regularization for overparameterized neural networks in the supervised lazy-training regime. We study denoising score matching in a vector-valued reproducing kernel Hilbert space with an inner-product kernel. In the proportional high-dimensional regime $n\asymp d$, we derive exact risk trajectories under gradient flow training. These trajectories exhibit three phases governed by qualitatively distinct estimators: a spectral estimator that generalizes, a pure-noise score with localized peaks that interpolate the training objective, and an empirical Bayes estimator that memorizes the data. We then analyze how these estimators combine along the reverse-time SDE and characterize the distribution of the resulting samples. The analysis reveals familiar mechanisms from supervised learning, including kernel linearization and self-induced regularization from the nonlinear part of the kernel, but also reveals a distinct phenomenology specific to generative modeling.
