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.

翻译：多维数据常伴随无法由正态分布充分描述的不确定性。本研究阐述了如何利用不确定性感知主成分分析（UAPCA）将此类分布投影至低维空间。我们提出使用高斯混合模型（GMMs）对多维分布进行建模，并从允许投影任意概率密度函数的通用公式推导出投影方法。相较于UAPCA映射，该方法得到的密度函数低维投影能展现更丰富的分布细节并实现更高保真度的表征。此外，我们支持在异质分布间引入用户自定义权重，从而可调节多维分布的重要性程度。通过对比本方法与UAPCA获得的低维空间分布与基于样本的投影结果，我们对所提方法进行了系统性评估。