Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise

• URL: http://arxiv.org/abs/2006.00402v2
• Date: Mon, 25 Jan 2021 20:58:51 GMT
• Title: Doubly-Stochastic Normalization of the Gaussian Kernel is Robust to Heteroskedastic Noise
• Authors: Boris Landa, Ronald R.Coifman, Yuval Kluger
• Abstract summary: We show that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal is robust to heteroskedastic noise. We provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity.
• Score: 3.5429774642987915
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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