**Title: Precise Stiefel Optimization for Probabilistic PLS: Closed-Form Updates, Error Bounds, and Calibrated Uncertainty

Abstract:

Probabilistic partial least squares (PPLS) serves as a primary likelihood-driven approach for two-view learning, particularly when the objective is to obtain interpretable latent factors alongside calibrated uncertainty estimates. While Bouhaddani et al. (2018) provided an identifiable parameterization, current fitting procedures remain hindered by two significant practical challenges: the coupling of noise and signal during joint EM/ECM updates, and the complex management of orthogonality constraints. Addressing these issues, we introduce a comprehensive framework that integrates noise pre-estimation, constrained likelihood optimization, and prediction calibration into a single pipeline, adhering to a fixed-noise scalar-likelihood protocol.

Our methodology involves estimating observation noise from the subspace associated with low eigenvalues and enforcing orthogonality via exact optimization on the Stiefel manifold. We demonstrate that this noise-subspace estimator achieves a leading finite-sample rate that is independent of signal strength, effectively matching a minimax lower bound. In contrast, a full-spectrum noise estimator introduces deterministic bias under the same model conditions. Furthermore, we adapt the framework for sub-Gaussian environments through optional Gaussianization and derive closed-form standard errors using a block-structured Fisher information analysis.

Empirical evaluations across synthetic high-noise scenarios and two multi-omics benchmarks—TCGA-BRCA and PBMC CITE-seq—highlight the method’s effectiveness. The approach yields near-nominal coverage without the need for post-hoc recalibration. On the TCGA-BRCA dataset, it attains Ridge-level point accuracy at rank $r=3$, while outperforming or matching PO2PLS in cross-view prediction tasks. Additionally, it offers native calibrated uncertainty and enhances the stability of parameter recovery.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC