Joint estimation of high-dimensional spiked covariance matrices via a partially shared subspace
Statistical analysis of high-dimensional data is often hampered by limited sample sizes, yet auxiliary datasets from related sources are often readily available. When two such datasets share part of their covariance structure, but not all of it, exploiting the shared part can substantially improve estimation. We propose a spiked covariance model that explicitly captures this partial sharing: two datasets share a subspace of unknown rank and arbitrary position in the spectrum, while each retains its own distinct spiked directions. The model treats the two datasets symmetrically and strictly generalizes existing models for shared covariance structure. We develop a complete estimation procedure that includes joint estimation of the shared subspace and its rank, a closed-form pooling weight for combining the two datasets, and asymptotic guarantees derived from random matrix theory in the proportional-growth regime. The framework also resolves a gap in contrastive dimension reduction by providing a principled estimator for high-dimensional settings. We illustrate the methodology on portfolio construction during the early COVID-19 pandemic and on contrastive analysis of brain tumor gene expression.