Optimal multi-task PCA beyond the bounded signal-to-noise regime

Develop multi-task PCA procedures that achieve optimality guarantees in regimes broader than bounded signal-to-noise ratios, including procedures based on dimension-free trimmed-mean-type covariance estimators within the two-stage multi-task learning framework.

Background

The paper’s depth-based procedure, DBMTL-PCA, is shown to attain minimax optimality only in a bounded signal-to-noise ratio regime. The stated limitation arises because its theoretical error bound depends on the ambient dimension rather than a dimension-free quantity such as the effective rank.

The authors note that trimmed-mean-type covariance estimators have recently achieved dimension-free bounds in single-task learning. Extending such ideas to the paper’s two-stage multi-task PCA framework is therefore identified as a concrete unresolved direction, with the goal of obtaining stronger optimality guarantees beyond the bounded signal-to-noise setting.