On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants
Effective discriminant dimensions can now exceed classical single-label limits, proven algebraically and statistically.
A unified theoretical analysis of orthogonal multilabel Fisher discriminants characterizes the between-class scatter matrix rank, showing effective dimensionality can exceed C-1. The four Fisher objectives are proven equivalent under the \(W^\top S_t^{ML} W = I_r\) constraint, with divergence characterized under the Stiefel constraint. Near-minimax-optimal subspace estimation error bounds matching within logarithmic and \(k_{\max}\) factors are derived. Numerical validation on synthetic data confirms the theory.
- Effective discriminant dimensionality can exceed C-1 for multilabel problems, breaking the classical single-label limitation.
- All four Fisher discriminant objectives are equivalent under the orthogonal constraint W^T S_t^{ML} W = I_r.
- Subspace estimation error bounds are near-minimax-optimal, matching lower bounds up to logarithmic factors and k_max.
Why It Matters
Provides rigorous theoretical foundation and optimality guarantees for multilabel classification, enabling more efficient and reliable dimensionality reduction.