Multilabel Fisher Discriminants: Algebraic Structure and Statistical Guarantees

A novel theoretical exploration of Linear Discriminant Analysis (LDA) incorporating multilabel scatter matrices alongside Stiefel orthogonality constraints has been introduced. This research defines the rank of the multilabel between-class scatter matrix, indicating that the effective discriminant dimensionality can surpass the traditional single-label limit of C-1. It formulates a multilabel variance partition and demonstrates the equivalence of four Fisher objectives under the constraint W^T S_t^{ML} W = I_r, while also detailing divergence under the Stiefel condition. Additionally, a two-sided bound on label-distance preservation that connects projected distances to Hamming distances in label space is established. On the statistical front, a finite-sample error bound for subspace estimation is derived under sub-Gaussian assumptions. This study contributes significantly to both algebraic and statistical aspects of multilabel classification.