Representation Learning with Blockwise Missingness and Signal Heterogeneity

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.

💡 Research Summary

The paper tackles a pervasive problem in multi‑source data integration: simultaneously dealing with structured blockwise missingness (different data sources observe only subsets of features) and blockwise signal heterogeneity (the strength of the latent signal varies across groups and feature blocks). Existing methods either ignore group‑specific blocks, rely solely on the shared feature block, or align group‑specific embeddings directly; all of these approaches break down when the shared block carries weak signal while some group‑specific blocks are strong.

To overcome this, the authors propose Anchor Projected Principal Component Analysis (APPCA). APPCA proceeds in two stages. In the first stage, for each subject group g, all observed feature blocks for that group are concatenated and a low‑rank PCA is performed, yielding a robust estimate of the group‑specific column subspace U_g (the column space of the subject embeddings Θ restricted to that group). By using every available block, the subspace estimate remains accurate even if some blocks have low signal‑to‑noise ratios.

In the second stage, a set of “anchor” features that are observed in multiple groups (typically the intersection of feature blocks) is projected onto each estimated subspace: the matrix P_{U_g} X_{·,V_anchor} is computed for every g. These projected anchor matrices are then stacked and a global PCA is applied. This projection dramatically reduces the effective noise level because high‑dimensional noise is confined to the low‑dimensional subspaces, providing a strong denoising effect. Crucially, the method does not require direct matching of noisy anchor embeddings across groups; instead it aligns the groups through the subspaces themselves.

The theoretical contribution is twofold. First, the authors develop a novel “spectral slicing” perturbation analysis for the group‑wise subspace recovery. Unlike classic Davis‑Kahan bounds, the resulting error bound does not depend on the condition number of the subject embedding matrix Θ, but only on the smallest singular value of the integrated feature matrix Φ. This makes the bound robust to weak subject signals. Second, they show that after projection, the effective signal‑to‑noise ratio of the anchor block is amplified, allowing the global PCA to achieve an estimation error of order O(p^{-1/2}) regardless of the weak shared‑block signal parameter β. In other words, the overall error depends solely on the strength of the combined feature blocks, not on the weakest component.

Empirically, the paper evaluates APPCA on synthetic data with 2×3 and 3×3 block patterns, varying the signal strength parameters β (shared block) and α (group‑specific subject signals). APPCA consistently outperforms baselines that (i) use only the shared block, (ii) perform a two‑step embedding alignment without projection, (iii) employ recent methods such as Cluster‑Quilting and Chain‑linked Multiple Matrix Integration (CMMI). When β is small (weak shared signal), the baselines’ error rates degrade to O(p^{-β/2}), while APPCA maintains the optimal O(p^{-1/2}) rate. The authors also demonstrate the method on multimodal single‑cell sequencing data (scRNA‑seq, scATAC‑seq, CITE‑seq). In this real‑world setting, feature modalities have markedly different signal strengths, yet APPCA yields a unified low‑dimensional embedding that improves cell‑type clustering (higher ARI) and downstream transfer‑learning tasks compared with existing integration pipelines.

Finally, the authors extend APPCA to the most general missingness patterns where no single feature block is shared across all groups. By constructing overlapping “super‑groups” and applying APPCA sequentially, they create a double‑anchor chain‑linking procedure that propagates alignment across the entire dataset while preserving robustness to heterogeneous signals. The same spectral‑slicing error analysis applies to each link, guaranteeing overall consistency.

In summary, APPCA introduces a principled, theoretically grounded framework that (1) leverages all observed data blocks for robust subspace recovery, (2) uses anchor projection to denoise and align groups without relying on weak shared signals, and (3) provides error bounds that are independent of subject‑signal conditioning. This makes it a powerful tool for a wide range of applications such as federated electronic health records, multimodal genomics, and any scenario where data are fragmented across sources with uneven signal quality.