Similarity Metrics: Chebyshev Coulomb Force and Resultant Force for High-Dimensional Data
Jianying Liu, Chaowei Zhang, Min Zhang, Xiao Qin, Jifu Zhang · ACM Transactions on Knowledge Discovery from Data · 2025
The similarity metric has garnered widespread attention thanks to its potential applications in the fields of data mining, machine learning, and so on. Due to the interference of “distance concentration” caused by “Curse of dimensionality,” however, existing similarity metrics are inadequate in high-dimensional data analysis. In this study, we propose two innovative similarity metrics—Chebyshev Coulomb force and Chebyshev Coulomb resultant force—anchored on Chebyshev p-norms. In the initial phase, we eliminate dependency relationships among attributes by applying a metric matrix—and the theoretical analysis reveals that the Chebyshev p-norms is capable of mitigating the effect of “distance concentration” among high-dimensional data objects. Next, we devise two similarity metrics—Chebyshev Coulomb force and Chebyshev Coulomb resultant force—by adopting the metric matrix and Chebyshev p-norms. Chebyshev Coulomb force and Chebyshev Coulomb resultant force, being effective in characterizing the similarity among data objects, quantify the deviation of data objects from their respective dataset centers. Additionally, the two metrics alleviate the interference of “distance concentration.” Importantly, the discrepancy of data objects in attribute dimensions is captured by Chebyshev Coulomb force vector, rendering the similarity metric interpretable. By utilizing the UCI dataset, the experimental validation demonstrates the superiority of our similarity metrics, confirming their efficacy in mitigating the interference of “distance concentration.” Compared with the existing similarity metric approaches, the AUC index of outlier detection shows an average improvement of 8.18%—and the ARI, NMI, and F_score indices of clustering are revamped by averages 6.56%, 6.87%, and 6.01%, respectively.