The success of machine learning models relies heavily on effectively representing high-dimensional data. However, ensuring data representations capture human-understandable concepts remains difficult, often requiring the incorporation of prior knowledge and decomposition of data into multiple subspaces. Traditional linear methods fall short in modeling more than one space, while more expressive deep learning approaches lack interpretability. Here, we introduce Supervised Independent Subspace Principal Component Analysis ($\texttt{sisPCA}$), a PCA extension designed for multi-subspace learning. Leveraging the Hilbert-Schmidt Independence Criterion (HSIC), $\texttt{sisPCA}$ incorporates supervision and simultaneously ensures subspace disentanglement. We demonstrate $\texttt{sisPCA}$'s connections with autoencoders and regularized linear regression and showcase its ability to identify and separate hidden data structures through extensive applications, including breast cancer diagnosis from image features, learning aging-associated DNA methylation changes, and single-cell analysis of malaria infection. Our results reveal distinct functional pathways associated with malaria colonization, underscoring the essentiality of explainable representation in high-dimensional data analysis.

翻译：机器学习模型的成功在很大程度上依赖于对高维数据的有效表示。然而，确保数据表示能够捕捉人类可理解的概念仍然具有挑战性，通常需要融入先验知识并将数据分解到多个子空间。传统线性方法在建模多于一个空间时存在不足，而表达能力更强的深度学习方法则缺乏可解释性。本文提出监督独立子空间主成分分析（$\texttt{sisPCA}$），这是一种专为多子空间学习设计的主成分分析扩展方法。通过利用希尔伯特-施密特独立性准则（HSIC），$\texttt{sisPCA}$ 融入了监督信息，并同时确保子空间的解耦。我们论证了 $\texttt{sisPCA}$ 与自编码器及正则化线性回归的联系，并通过广泛的应用展示了其识别和分离隐藏数据结构的能力，这些应用包括基于图像特征的乳腺癌诊断、学习与衰老相关的DNA甲基化变化以及疟疾感染的单细胞分析。我们的结果揭示了与疟疾定植相关的不同功能通路，凸显了可解释表示在高维数据分析中的重要性。