用高斯混合模型改进主成分分析，更好处理不确定数据的低维投影。
Uncertainty-Aware PCA for Arbitrarily Distributed Data Modeled by Gaussian Mixture Models
- 基于高斯混合模型建模多维不确定数据，实现任意分布的低维投影。
- 新方法比传统方法保留更多分布细节，投影更忠实于原始数据。
- 支持用户自定义权重，灵活调整不同分布的重要性，适合数据分析场景。
多维数据常伴随不确定性，且难以用正态分布描述。本文提出不确定性感知主成分分析（UAPCA）的新方法，采用高斯混合模型（GMM）建模多维分布，并推导出可对任意概率密度函数进行投影的一般公式。与传统UAPCA相比，该方法在低维空间中对密度的投影能展现更多分布细节，更真实地表示原始分布。此外，方法支持用户自定义各分量的权重，以调节不同分布的重要性。通过与基于样本的投影结果对比，验证了本方法在低维表示上的优越性。
原文摘要 · Abstract (English)
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.
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