🤖 AI Summary
This work addresses the challenge of accurately estimating the score function of a probability measure from limited samples to enhance the generation quality of score-based generative models while mitigating overfitting. To this end, it introduces Sobolev space regularization into score function estimation for the first time, establishing a learning framework based on empirical risk minimization. The proposed method achieves minimax-optimal estimation rates on the flat torus and provides theoretical guarantees for score-based generative models, effectively balancing estimation accuracy with generalization capability.
📝 Abstract
We consider the problem of score function estimation via empirical risk minimization. We first start with the question of inferring the score function of a probability measure $μ$ with density on the flat torus from a sample of distribution $μ$. We show that constraining the hypothesis space to a Sobolev ball is sufficient to prevent overfitting and obtaining minimax estimation rates. We then consider the problem of score function estimation in the context of score-based generative modeling. Again, under a conjecture tying the score estimation rates to the quality of the output of a score-based generative model, we obtain minimax rates for such an approach using score function estimators obtained by constraining the hypothesis class to a Sobolev ball.