Fixed Point Theorems in G-Metric Vector Spaces and their applications in Machine Learning and Optimization Algorithms

This research aims to develop a novel mathematical formulation of fixed point theorems in G-metric vector spaces and explore its applications in machine learning and optimization algorithms. The study begins by defining a complete G-metric space and establishing a generalized contraction condition for an operator representing iterative updates in optimization processes. Utilizing gradient descent with regularization as a case study, a numerical example is presented to validate the proposed formulation. The methodology involves iterative calculations demonstrating the convergence of a sequence of parameters toward a fixed point, highlighting the effectiveness of the auxiliary function in promoting sparsity while accommodating the non-Euclidean characteristics of high-dimensional data. The main results reveal that the proposed framework not only enhances the convergence properties of iterative algorithms but also aligns well with contemporary regularization techniques in machine learning. The conclusions drawn emphasize the significance of integrating generalized G-metric spaces and auxiliary functions in fixed point theories, suggesting that this approach provides a robust foundation for developing adaptive optimization algorithms. The findings indicate substantial implications for future research, highlighting the need for broader exploration of G-metric spaces and their applicability in various machine learning paradigms to address complex data challenges.