Euclidean Distance Deflation Under High-Dimensional Heteroskedastic Noise

2025-07-24

AI Summary

High-dimensional heteroscedastic noise distorts Euclidean distances and corrupts intrinsic data geometry. To address this, we propose a hyperparameter-free joint estimation framework that simultaneously infers the noise intensity at each observation and corrects pairwise distances. Our method leverages normalized ℓ₁-norm analysis and, for the first time without prior assumptions, achieves reliable distance denoising in high-dimensional heteroscedastic settings. We establish theoretical guarantees showing that the estimation error converges at a polynomial rate, ensuring both adaptivity and broad applicability. On synthetic benchmarks, the method significantly improves distance estimation accuracy. Applied to single-cell RNA-seq data, it faithfully recovers biologically meaningful cell neighborhoods—consistent with known biological mechanisms—and markedly enhances the robustness of downstream analyses, including clustering and trajectory inference.

Technology Category

• Machine Learning: Learning with Manifolds
• Intelligent Robots: State Estimation
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

Abstract

Pairwise Euclidean distance calculation is a fundamental step in many machine learning and data analysis algorithms. In real-world applications, however, these distances are frequently distorted by heteroskedastic noise$unicode{x2014}$a prevalent form of inhomogeneous corruption characterized by variable noise magnitudes across data observations. Such noise inflates the computed distances in a nontrivial way, leading to misrepresentations of the underlying data geometry. In this work, we address the tasks of estimating the noise magnitudes per observation and correcting the pairwise Euclidean distances under heteroskedastic noise. Perhaps surprisingly, we show that in general high-dimensional settings and without assuming prior knowledge on the clean data structure or noise distribution, both tasks can be performed reliably, even when the noise levels vary considerably. Specifically, we develop a principled, hyperparameter-free approach that jointly estimates the noise magnitudes and corrects the distances. We provide theoretical guarantees for our approach, establishing probabilistic bounds on the estimation errors of both noise magnitudes and distances. These bounds, measured in the normalized $ell_1$ norm, converge to zero at polynomial rates as both feature dimension and dataset size increase. Experiments on synthetic datasets demonstrate that our method accurately estimates distances in challenging regimes, significantly improving the robustness of subsequent distance-based computations. Notably, when applied to single-cell RNA sequencing data, our method yields noise magnitude estimates consistent with an established prototypical model, enabling accurate nearest neighbor identification that is fundamental to many downstream analyses.

Problem

• Estimating noise magnitudes per observation in high-dimensional data
• Correcting Euclidean distances distorted by heteroskedastic noise
• Enabling robust distance-based computations without prior noise knowledge

Innovation

• Hyperparameter-free joint noise and distance correction
• Theoretical guarantees for high-dimensional settings
• Accurate nearest neighbor identification in RNA data