Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling

TL;DR

This paper introduces a doubly stochastic normalization method for Gaussian kernels that enhances robustness in high-dimensional noisy data, enabling more accurate inference of manifold density, geometry, and related structures.

Contribution

The authors develop a novel doubly stochastic normalization approach that improves robustness and accuracy in manifold inference under high-dimensional, heteroskedastic noise conditions.

Findings

Doubly stochastic affinity matrices concentrate around population forms.

The method outperforms standard kernel density estimators under heteroskedasticity.

Robust graph Laplacian normalizations better approximate manifold Laplacians in noisy data.

Abstract

The Gaussian kernel and its traditional normalizations (e.g., row-stochastic) are popular approaches for assessing similarities between data points. Yet, they can be inaccurate under high-dimensional noise, especially if the noise magnitude varies considerably across the data, e.g., under heteroskedasticity or outliers. In this work, we investigate a more robust alternative -- the doubly stochastic normalization of the Gaussian kernel. We consider a setting where points are sampled from an unknown density on a low-dimensional manifold embedded in high-dimensional space and corrupted by possibly strong, non-identically distributed, sub-Gaussian noise. We establish that the doubly stochastic affinity matrix and its scaling factors concentrate around certain population forms, and provide corresponding finite-sample probabilistic error bounds. We then utilize these results to develop several tools for robust inference under general high-dimensional noise. First, we derive a robust density estimator that reliably infers the underlying sampling density and can substantially outperform the standard kernel density estimator under heteroskedasticity and outliers. Second, we obtain estimators for the pointwise noise magnitudes, the pointwise signal magnitudes, and the pairwise Euclidean distances between clean data points. Lastly, we derive robust graph Laplacian normalizations that accurately approximate various manifold Laplacians, including the Laplace Beltrami operator, improving over traditional normalizations in noisy settings. We exemplify our results in simulations and on real single-cell RNA-sequencing data. For the latter, we show that in contrast to traditional methods, our approach is robust to variability in technical noise levels across cell types.