Title: Pairwise Worst-Case Ratio Analysis for Discriminative Dimensionality Reduction via Minorization-Maximization
Primary area: optimization
Keywords: Discriminative Dimensionality Reduction, Fractional Quadratic Programming, Pairwise Worst-Case Ratio Analysis (PWCRA), Max-Min Optimization

Abstract: In this paper, we investigate a novel discriminative dimensionality reduction method based on maximizing the minimum pairwise ratio of between-class to within-class scatter. This objective function enhances class separability by providing critical, adaptive control over the variance within each class pair. The resulting max-min fractional programming problem is non-convex and notoriously challenging to solve. Our key contribution is a provably convergent, two-level iterative algorithm, termed GDMM-QF (generalized Dinkelbach-minorization-maximization for quadratic fractional programs), to find a high-quality solution. The outer loop employs a generalized Dinkelbach-type procedure to transform the fractional program into an equivalent sequence of subtractive-form max-min subproblems. For the inner loop, we develop an efficient minorization-maximization (MM) algorithm that tackles the non-convex subproblem by iteratively solving a simple quadratic program (QP), which we derive from the dual of a convex surrogate. The proposed GDMM-QF framework is computationally efficient, guaranteed to converge, and requires no hyperparameter tuning. Experiments on multiple benchmark datasets confirm the superiority of our method in learning discriminative projections, consistently achieving lower classification error than state-of-the-art alternatives.