Comparative framework for analyzing distance metrics in high-dimensional spaces
DOI:
https://doi.org/10.30837/2522-9818.2025.1.143Keywords:
distance metrics; high-dimensional spaces; metric comparison; norm; norm; data analysis; machine learning.Abstract
Subject of the research – developing a comprehensive framework to measure and analyze the relationships between different distance metrics in high-dimensional spaces. Aim of the research – to create a comparative framework that quantifies the "distance" between various distance metrics in high-dimensional settings. This framework aims to provide deeper insights into the interrelationships of these metrics and to guide practitioners in selecting the most appropriate metric for specific data analysis tasks. The research tasks include a theoretical formulation of methods to measure the "distance between distances", enabling a systematic comparison of different metrics. We conduct a thorough analysis of how these relationships evolve with increasing dimensionality. This involves developing mathematical models and employing visualization techniques to illustrate and interpret the relationships between metrics like the Manhattan distance, Euclidean distance, and others in high-dimensional spaces. A series of experiments are conducted on synthetic datasets to validate the theoretical findings and demonstrate the practical utility of the proposed framework. These datasets are carefully selected to cover a wide range of dimensionalities and data characteristics, ensuring a comprehensive evaluation of the framework's effectiveness. The methodology includes statistical analyses and visualization methods such as multidimensional scaling and heatmaps to represent the relationships between distance metrics clearly. The findings of the research are significant, revealing that the relationships between different distance metrics change notably as dimensionality increases. The results show patterns of convergence or divergence among certain metrics, providing valuable insights into their behavior in high-dimensional spaces. These insights are crucial for improving the accuracy and efficiency of data analysis techniques that rely on distance computations. The conclusions indicate that the proposed framework successfully quantifies the relationships between various distance metrics in high-dimensional spaces. By enhancing the understanding of how these metrics relate to one another, the research offers a valuable tool for selecting appropriate distance measures in high-dimensional data analysis. This contributes to more accurate and efficient analytical processes across various fields, including machine learning, data mining, and pattern recognition.
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