Synergistic eigenanalysis of covariance and Hessian matrices for enhanced binary classification

Covariance and Hessian matrices have been analyzed separately in the literature for classification problems. However, integrating these matrices has the potential to enhance their combined power in improving classification performance. We present a novel approach that combines the eigenanalysis of a covariance matrix evaluated on a training set with a Hessian matrix evaluated on a deep learning model to achieve optimal class separability in binary classification tasks. Our approach is substantiated by formal proofs that establish its capability to maximize between-class mean distance and minimize within-class variances. By projecting data into the combined space of the most relevant eigendirections from both matrices, we achieve optimal class separability as per the linear discriminant analysis (LDA) criteria. Empirical validation across neural and health datasets consistently supports our theoretical framework and demonstrates that our method outperforms established methods. Our method stands out by addressing both LDA criteria, unlike PCA and the Hessian method, which predominantly emphasize one criterion each. This comprehensive approach captures intricate patterns and relationships, enhancing classification performance. Furthermore, through the utilization of both LDA criteria, our method outperforms LDA itself by leveraging higher-dimensional feature spaces, in accordance with Cover's theorem, which favors linear separability in higher dimensions. Our method also surpasses kernel-based methods and manifold learning techniques in performance. Additionally, our approach sheds light on complex DNN decision-making, rendering them comprehensible within a 2D space.