Inference for PCA without an eigengap (arbitrary singular subspaces)

Establish valid statistical inference for principal component analysis when the data matrix X_n^T X_n has repeated eigenvalues (i.e., without the global relative eigengap), by developing asymptotic distributions and confidence procedures for singular values associated with arbitrary singular subspaces, thereby relaxing the distinct-eigenvalues requirement used in the framework.

Background

The paper’s PCA inference results require Condition B (a global relative eigengap), which ensures all eigenvalues are distinct. This facilitates differentiability of eigenvalue/eigenvector maps and enables delta-method arguments for asymptotics.

The authors note that relaxing this assumption to allow repeated eigenvalues would necessitate inference for targets associated with singular subspaces rather than individual eigenvectors. They explicitly flag this extension as future work, indicating that their current theory does not cover this important generalization.