QR-FOLDA: A Fast Orthogonal Linear Discriminant Analysis Based on QR Decomposition

Orthogonal Linear Discriminant Analysis (OLDA) has been widely studied for dimensionality reduction, as the orthogonality constraints on its projection matrix enable more effective elimination of redundant features compared to conventional Linear Discriminant Analysis (LDA). However, most existing methods for solving OLDA rely on iterative optimization to sequentially construct orthogonal components, incurring massive repeated high-cost matrix operations and thus leading to substantial computational inefficiency. To address this limitation, we propose QR decomposition-based Fast OLDA (QR-FOLDA), a method built upon a theoretical result (Theorem 1) established in this work: the optimal solutions of LDA remain valid under any full-rank linear transformation. By leveraging this property, QR-FOLDA applies QR decomposition directly to the optimal LDA solution, thereby enforcing orthogonality while avoiding the repeated high-cost matrix operations typically involved in iterative optimization procedures. Experimental evaluations conducted on nine real-world datasets across various domains show that QR-FOLDA not only achieves substantial improvements in computational efficiency compared to existing OLDA methods but also delivers superior classification performance. These findings position QR-FOLDA as a theoretically sound and practically efficient solution for orthogonal discriminant analysis.