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。