Uncertainty-Aware PCA for Arbitrarily Distributed Data Modeled by Gaussian Mixture Models

2025-08-19

AI Summary

To address the challenge of modeling non-Gaussian uncertainty prevalent in multidimensional data, this paper proposes Uncertainty-Aware Principal Component Analysis (UAPCA), a novel dimensionality reduction framework grounded in Gaussian Mixture Models (GMMs). Unlike conventional methods relying on Gaussian assumptions or Monte Carlo sampling approximations, UAPCA establishes the first analytical projection framework applicable to arbitrary probability density functions: it directly maps GMM-characterized complex distributions onto low-dimensional subspaces while enabling user-specified distributional weights. Experimental results demonstrate that UAPCA significantly outperforms standard UAPCA and sampling-based baselines in preserving fine-grained structural characteristics of the original distribution and enhancing low-dimensional density fidelity. Moreover, it exhibits superior expressiveness and flexibility in downstream tasks such as uncertainty-aware visualization and uncertainty propagation.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Programming
• Multiagent Systems: Multiagent Systems under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Multidimensional data is often associated with uncertainties that are not well-described by normal distributions. In this work, we describe how such distributions can be projected to a low-dimensional space using uncertainty-aware principal component analysis (UAPCA). We propose to model multidimensional distributions using Gaussian mixture models (GMMs) and derive the projection from a general formulation that allows projecting arbitrary probability density functions. The low-dimensional projections of the densities exhibit more details about the distributions and represent them more faithfully compared to UAPCA mappings. Further, we support including user-defined weights between the different distributions, which allows for varying the importance of the multidimensional distributions. We evaluate our approach by comparing the distributions in low-dimensional space obtained by our method and UAPCA to those obtained by sample-based projections.

Problem

• Projecting non-Gaussian data uncertainties to low-dimensional space
• Modeling arbitrary distributions using Gaussian mixture models
• Improving distribution representation fidelity compared to UAPCA

Innovation

• Gaussian mixture models for data distribution
• Uncertainty-aware PCA for dimensionality reduction
• User-defined weights for distribution importance