用神经场直接建模流形上的概率密度，解决高维数据估计难题。
NeuroPMD: Neural Fields for Density Estimation on Product Manifolds
- 通过神经网络直接参数化密度函数，结合流形微分算子设计正则项
- 在脑结构连接数据上表现优于传统核方法与普通神经网络
- 适合高维流形空间的概率密度估计，尤其适用于医学影像分析
我们提出一种新型深度神经网络方法，用于在乘积黎曼流形域上进行密度估计。该方法直接由网络参数化未知密度函数，并采用带惩罚项的最大似然框架进行训练，惩罚项由流形微分算子构成。网络架构与估计算法经过精心设计，以应对高维乘积流形域的挑战，有效缓解传统核方法与基函数展开估计器面临的维度诅咒问题，同时克服非专用神经网络方法存在的收敛困难。大量模拟实验及一个真实世界应用——脑结构连接数据——表明该方法显著优于现有竞争方法。
原文摘要 · Abstract (English)
We propose a novel deep neural network methodology for density estimation on product Riemannian manifold domains. In our approach, the network directly parameterizes the unknown density function and is trained using a penalized maximum likelihood framework, with a penalty term formed using manifold differential operators. The network architecture and estimation algorithm are carefully designed to handle the challenges of high-dimensional product manifold domains, effectively mitigating the curse of dimensionality that limits traditional kernel and basis expansion estimators, as well as overcoming the convergence issues encountered by non-specialized neural network methods. Extensive simulations and a real-world application to brain structural connectivity data highlight the clear advantages of our method over the competing alternatives.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。