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.
翻译:我们引入了一种叫做扩散曲线的点球数据局部曲线的新的内在测量方法。 我们的测量方法使用了包括数据扩散操作员在内的扩散地图框架,以根据从数据的一个或一个区域开始的随机行走的懒惰性来构建点云数据和定义本地曲线。 我们显示,这种偏移与里曼尼地理测量的量比较结果直接相关。 然后我们使用基于点云数据扩散图的神经网络估计,将这个弧曲线概念扩展至整个二次形。 我们显示了对玩具数据、单细胞数据以及神经网络损失图景的赫森本地矩阵的估计的应用。