揭示多标签Fisher判别分析的谱结构，证明其目标函数等价性与最优降维率。
On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants
- 建立多标签类间散度矩阵的秩与方差分解理论，突破单标签维数上限。
- 在正交约束下四类Fisher目标函数等价，且投影距离与标签汉明距离有严格保距关系。
- 给出近最优的有限样本子空间估计误差界，适合理论研究者与高维多标签学习场景。
本文对具有多重标签散度矩阵形式与Stiefel正交约束的线性判别分析进行统一理论分析。代数层面，刻画了多标签类间散度矩阵的秩，表明有效判别维度可严格超过经典单标签的 $C-1$ 上限；建立了多标签方差分解，证明在 $W^ op S_t^{ML} W = I_r$ 约束下四个Fisher目标等价，而正交约束下存在差异；并证明了投影距离与标签空间汉明距离之间的双侧保距界。统计层面，给出了在亚高斯噪声下子空间估计误差的有限样本上界 $O(k_{ ext{max}} rac{\\
原文摘要 · Abstract (English)
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 $W^\top S_t^{ML} W = I_r$ 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 $O(k_{\max}\sqrt{d\log d/n}/gap_r)$ bound on the subspace estimation error under sub-Gaussian noise with a matching $Ω(σ^2 d/(n\,gap_r))$ minimax lower bound, establishing a near-minimax-optimal rate (matching up to logarithmic and $k_{\max}$ 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 ($k_{\max}$, $κ(S_t^{ML})$, $\|Γ/n\|_2$, $Δ_r$) 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.
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