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