Nesterov Accelerated Gradient Descent for Optimizing Fast Harmonic Mean Linear Discriminant Analysis

Dimension reduction algorithms have become widespread in data science due to the prevalence of High-Dimensional Data (HDD).In recent years, many versions of Linear Discriminant Analysis (LDA) have been developed for dimensionality reduction.Among them, the Fast Harmonic mean-based LDA (FHLDA) and FHLDA-pairwise (FHLDAp) algorithms reduce HDD by adopting joint diagonalization based on Taylor expansion to generate discriminants.However, the Stiefel manifold gradient scheme in these algorithms involves many matrix multiplications, leading to high computational time complexity (( 2 )).Thus, this manuscript proposes an Accelerated Optimization (AO) approach for FHLDA and FHLDAp algorithms to reduce the complexity of the Stiefel manifold gradient scheme to (√) .A Nesterov accelerated gradient descent scheme is introduced to optimize functions on the Stiefel manifold by constructing a sequence of proximal points satisfying manifold constraints.This achieves asymptotically optimal error for L-smooth convex, as well as L-smooth and -strongly convex functions, provided step size satisfies the Lipschitz smoothness condition.So, it is ensured to converge and achieve an accelerated rate after the solution is nearer to the local.After applying this scheme, joint diagonalization via Taylor expansion recovers the discriminant vector from the manifold.Experimental results demonstrate that the AOFHLDA and AOFHLDAp algorithms outperform LDA, FHLDA, and FHLDAp on both single and multi-label datasets, achieving significant accuracy improvements.Specifically, AOFHLDA improves accuracy by 17.87%, 14.14%, 14.65%, and 15.28% on the PIE, UMIST, Barcelona, and MediaMill datasets, respectively.Similarly, AOFHLDAp improves accuracy by 19.32%, 15.13%, 15.61%, and 16.44% on the PIE, UMIST, Barcelona, and MediaMill datasets, respectively.