This paper introduces a novel, non-deterministic method for embedding data in low-dimensional Euclidean space based on computing realizations of a Gaussian process depending on the geometry of the data. This type of embedding first appeared in (Adler et al, 2018) as a theoretical model for a generic manifold in high dimensions. In particular, we take the covariance function of the Gaussian process to be the heat kernel, and computing the embedding amounts to sketching a matrix representing the heat kernel. The Karhunen-Lo\`eve expansion reveals that the straight-line distances in the embedding approximate the diffusion distance in a probabilistic sense, avoiding the need for sharp cutoffs and maintaining some of the smaller-scale structure. Our method demonstrates further advantage in its robustness to outliers. We justify the approach with both theory and experiments.

引入一种新的非确定性方法，将数据嵌入低维欧几里德空间，该方法基于依赖于数据几何的高斯过程的实现，通过计算高斯过程的实现来计算嵌入，其中高斯过程的协方差函数被取为热核函数，嵌入的直线距离以概率意义上逼近扩散距离，避免了对距离进行尖锐截断并保留了一定的较小尺度结构，此方法还具有对异常值的鲁棒性，并通过理论和实验证明了该方法的优势。

将热核进行草图化：使用高斯过程嵌入数据