具有随机特征得分和流形数据的扩散模型的渐近学习曲线
Asymptotic Learning Curves for Diffusion Models with Random Features Score and Manifold Data
AI总结：
研究扩散模型中去噪得分匹配的理论行为，当数据分布支持在低维流形上且得分用随机特征神经网络参数化时，推导高维极限下测试、训练和得分误差的渐近精确表达式，发现线性流形样本复杂度与内在维度线性相关，而非线性流形则削弱低维结构优势。
摘要：
我们研究了去噪得分匹配——与扩散模型相关的学习任务——的理论行为，当数据分布支持在低维流形上且得分使用随机特征神经网络参数化时。我们在高维极限下推导了测试、训练和得分误差的渐近精确表达式。我们的分析揭示，对于线性流形，学习得分函数所需的样本复杂度与流形的内在维度线性相关，而不是与环境维度线性相关。令人惊讶的是，一旦我们拥有非线性流形，低维结构的好处开始减弱。这些结果表明扩散模型可以从结构化数据中受益；然而，对特定类型结构的依赖是微妙而复杂的。
英文摘要：
We study the theoretical behavior of denoising score matching--the learning task associated to diffusion models--when the data distribution is supported on a low-dimensional manifold and the score is parameterized using a random feature neural network. We derive asymptotically exact expressions for the test, train, and score errors in the high-dimensional limit. Our analysis reveals that, for linear manifolds the sample complexity required to learn the score function scales linearly with the intrinsic dimension of the manifold, rather than with the ambient dimension. Perhaps surprisingly, the benefits of low-dimensional structure starts to diminish once we have a non-linear manifold. These results indicate that diffusion models can benefit from structured data; however, the dependence on the specific type of structure is subtle and intricate.
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