On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants

We provide a unified theoretical analysis of Linear Discriminant Analysis with simultaneous multilabel scatter matrix formulations and Stiefel orthogonality constraints. Our contributions span both algebraic structure and statistical guarantees. On the algebraic side, we characterize the rank of the multilabel between-class scatter matrix, showing that the effective discriminant dimensionality can strictly exceed the classical single-label bound of C-1; we establish a multilabel partition of variance and prove that all four Fisher objectives are equivalent under the W^top StML W = Ir constraint while characterizing their divergence under the Stiefel constraint; and we prove a two-sided label-distance preservation bound relating projected distances to Hamming distances in label space. On the statistical side, we establish a finite-sample O(kmax√dlog d/n/gapr) bound on the subspace estimation error under sub-Gaussian noise with a matching Omega(σ2 d/(n,gapr)) minimax lower bound, establishing a near-minimax-optimal rate (matching up to logarithmic and kmax factors) for multilabel discriminant subspace estimation. We further provide high-probability distance concentration, robustness guarantees under label interactions, and a regularization analysis preserving the spectral structure when d gg n. All results are verified numerically on synthetic data generated from the linear label-effect model, covering both the algebraic identities and the multilabel-specific quantities (kmax, kappa(StML), |Gamma/n|2, Deltar) that govern the statistical bounds. The numerical experiments are designed as a sanity check for the theorems rather than as an empirical benchmark; evaluation on real multilabel datasets is left to future work targeting application-oriented venues.