Doubly Stochastic Normalization of the Gaussian Kernel Is Robust to Heteroskedastic Noise

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When points reside Euclidean space, a widespread approach to form by Gaussian kernel with pairwise distances, and follow certain normalization (e.g., row-stochastic or its symmetric variant). We demonstrate that doubly stochastic zero main diagonal (i.e., no self-loops) robust heteroskedastic noise. That is, advantageous it automatically accounts for observations different noise variances. Specifically, we prove suitable high-dimensional setting where does not concentrate too much any particular direction resulting (doubly stochastic) noisy converges clean counterpart rate $m^{-1/2}$, $m$ ambient dimension. this result numerically show that, contrast, popular normalizations behave unfavorably under Furthermore, provide examples simulated experimental single-cell RNA sequence intrinsic heteroskedasticity, advantage exploratory analysis evident.