论文标题
扩散曲率用于估计高维数据中局部曲率的曲率
Diffusion Curvature for Estimating Local Curvature in High Dimensional Data
论文作者
论文摘要
我们在点云数据上引入了一种局部曲率的新固有度量，称为扩散曲率。我们的度量使用扩散图的框架，包括数据扩散操作员，构建点云数据，并根据从数据的点或区域开始的随机步行的懒惰定义局部曲率。我们表明，这种懒惰直接与Riemannian几何形状的体积比较结果有关。然后，我们使用基于点云数据扩散图的神经网络估计将此标态曲率概念扩展到整个二次形式。我们展示了对玩具数据，单细胞数据的估计以及估计神经网络损失景观局部Hessian矩阵的应用。
We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data, and on estimating local Hessian matrices of neural network loss landscapes.
