Unified representation learning for multi-source data integration faces two important challenges: blockwise missingness and blockwise signal heterogeneity. The former arises from sources observing different, yet potentially overlapping, feature sets, while the latter involves varying signal strengths across subject groups and feature sets. While existing methods perform well with fully observed data or uniform signal strength, their performance degenerates when these two challenges coincide, which is common in practice. To address this, we propose Anchor Projected Principal Component Analysis (APPCA), a general framework for representation learning with structured blockwise missingness that is robust to signal heterogeneity. APPCA first recovers robust group-specific column spaces using all observed feature sets, and then aligns them by projecting shared "anchor" features onto these subspaces before performing PCA. This projection step induces a significant denoising effect. We establish estimation error bounds for embedding reconstruction through a fine-grained perturbation analysis. In particular, using a novel spectral slicing technique, our bound eliminates the standard dependency on the signal strength of subject embeddings, relying instead solely on the signal strength of integrated feature sets. We validate the proposed method through extensive simulation studies and an application to multimodal single-cell sequencing data.

翻译：多源数据整合的统一表示学习面临两大挑战：块缺失与块信号异质性。前者源于不同数据源观测到可能重叠但非完全一致的特征集合，后者则涉及不同样本组与特征集合间信号强度的差异。现有方法在数据完全可观测或信号强度均匀时表现良好，但当这两种挑战同时出现时——这在实践中十分常见——其性能会显著下降。为解决这一问题，我们提出锚点投影主成分分析（APPCA），这是一个针对结构化块缺失的表示学习通用框架，对信号异质性具有鲁棒性。APPCA首先利用所有可观测特征集合恢复鲁棒的组特异性列空间，然后通过将共享的“锚点”特征投影到这些子空间中进行对齐，最后执行PCA。该投影步骤能产生显著的降噪效果。我们通过细粒度扰动分析建立了嵌入重构的估计误差界。特别地，借助新颖的谱切片技术，我们的误差界消除了传统方法对样本嵌入信号强度的依赖，转而仅依赖于整合后特征集合的信号强度。我们通过大量模拟研究及在多模态单细胞测序数据上的应用验证了所提方法的有效性。