ADVANCED LINEAR ALGEBRA AND MATHEMATICAL OPTIMIZATION TECHNIQUES FOR HIGH-DIMENSIONAL DATA ANALYSIS AND MACHINE LEARNING APPLICATIONS

Abstract

The rapid expansion of high-dimensional data in artificial intelligence, machine learning, and data science introduces significant challenges in computational efficiency, scalability, feature extraction, and optimization stability. Traditional dimensionality reduction and learning techniques often struggle to maintain accuracy and convergence when applied to large-scale or nonlinear datasets. This study proposes an advanced mathematical framework that integrates linear algebra-based representations with adaptive optimization strategies to improve the efficiency and reliability of high-dimensional data analysis. The framework combines sparse matrix representation, low-rank decomposition, tensor factorization, and spectral regularization to preserve data structure and reduce dimensional complexity.

An Adaptive Spectral Regularized Optimization (ASRO) mechanism is introduced to dynamically adjust learning behaviour, enhancing convergence stability and reducing overfitting. Experimental evaluation across benchmark datasets covering image recognition, biomedical analysis, financial prediction, and text classification demonstrates that the proposed framework outperforms conventional methods such as Principal Component Analysis (PCA), stochastic gradient descent, and standard matrix factorization techniques. On average, it achieves a 7.8% improvement in classification accuracy, 18.6% higher dimensionality reduction efficiency, 24.3% faster convergence, and a 31% reduction in training latency in distributed environments. These results confirm that integrating advanced linear algebra with adaptive optimization significantly enhances the scalability, stability, and predictive performance of machine learning models in high-dimensional settings.