标题： 基于扩散模型的可分解密度的非参数估计 显示英文标题

标题： Nonparametric estimation of a factorizable density using diffusion models

摘要： 近年来，扩散模型，更一般地讲，基于分数的深度生成模型，在各种应用中取得了显著的成功，包括图像和音频生成。 在本文中，我们将扩散模型视为一种非参数密度估计的隐式方法，并在统计框架内研究它们以分析其令人惊讶的性能。 高维统计推断中的一个关键挑战是利用数据中固有的低维结构来缓解维度灾难。 我们假设底层密度通过分解为低维成分表现出低维结构，这是贝叶斯网络和马尔可夫随机场等例子中的常见性质。 在适当的假设下，我们证明了从扩散模型构建的隐式密度估计器能够适应分解结构，并在总变分距离方面达到最小最大最优率。 在构建估计器时，我们设计了一种稀疏权重共享的神经网络架构，其中稀疏性和权重共享是卷积神经网络和循环神经网络等实际架构的关键特征。 显示英文摘要

摘要： In recent years, diffusion models, and more generally score-based deep generative models, have achieved remarkable success in various applications, including image and audio generation. In this paper, we view diffusion models as an implicit approach to nonparametric density estimation and study them within a statistical framework to analyze their surprising performance. A key challenge in high-dimensional statistical inference is leveraging low-dimensional structures inherent in the data to mitigate the curse of dimensionality. We assume that the underlying density exhibits a low-dimensional structure by factorizing into low-dimensional components, a property common in examples such as Bayesian networks and Markov random fields. Under suitable assumptions, we demonstrate that an implicit density estimator constructed from diffusion models adapts to the factorization structure and achieves the minimax optimal rate with respect to the total variation distance. In constructing the estimator, we design a sparse weight-sharing neural network architecture, where sparsity and weight-sharing are key features of practical architectures such as convolutional neural networks and recurrent neural networks.