基于双重随机归一化的流形密度与几何的鲁棒推断
Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling
AI总结：
本文研究高斯核的双重随机归一化，在高维异方差噪声下建立其集中性误差界，并据此提出鲁棒密度估计、噪声与信号幅度估计及图拉普拉斯归一化方法，在模拟和单细胞RNA测序数据中验证了其鲁棒性。
摘要：
高斯核及其传统归一化方式（例如行随机归一化）是评估数据点之间相似性的常用方法。然而，在高维噪声下，尤其是当噪声幅度在数据中变化很大时（例如在异方差或离群点情形下），这些方法可能不准确。本文研究了一种更鲁棒的替代方案——高斯核的双重随机归一化。我们考虑如下设置：数据点从嵌入在高维空间中的低维流形上的未知密度中采样，并受到可能较强、非同一分布、亚高斯噪声的污染。我们证明双重随机亲和矩阵及其缩放因子会围绕某些总体形式集中，并给出相应的有限样本概率误差界。随后，我们利用这些结果开发了若干用于一般高维噪声下鲁棒推断的工具。首先，我们推导出一个鲁棒密度估计器，它能够可靠地推断底层采样密度，并且在异方差和离群点情形下可以显著优于标准核密度估计器。其次，我们得到了逐点噪声幅度、逐点信号幅度以及干净数据点之间成对欧氏距离的估计器。最后，我们推导出鲁棒的图拉普拉斯归一化方法，能够准确近似各种流形拉普拉斯算子，包括Laplace Beltrami算子，在噪声环境中优于传统归一化方法。我们在模拟实验和真实单细胞RNA测序数据上验证了结果。对于后者，我们表明与传统方法相比，我们的方法对跨细胞类型的技术噪声水平变化具有鲁棒性。
英文摘要：
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.
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