International Journal on Science and Technology

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Hyperbolic-Valued Multi-Norms on Bicomplex Vector Spaces: Theory and Machine Learning Applications

Author(s) Neetu Singh
Country India
Abstract Background: Multi-norms represent the joint geometry across finite tuples, while bicomplex modules naturally decompose into two idempotent complex parts. The connection has not been much explored in the literature of statistical learning. Methods. A hyperbolic-valued multi-norm was defined component-wise, and decomposition, product-topology and completeness were explored. Pilot studies were conducted on the UCI Breast Cancer Wisconsin Diagnostic dataset (569 cases, 30 variables) using a scalarised, jointly regularised logistic classifier. Stratified 80:20 data split, 5-fold cross-validation, a fixed random seed (42) and controlled Gaussian perturbation were used. Results. The classifier exhibited 0.974 test accuracy, 0.979 test F1, and an area under the ROC curve (AUC) of 0.996. Cross-validation accuracy was 0.982 ± 0.006. The dual-component baseline reached the maximum clean test accuracy of 0.982. The hyperbolic model retained 0.913 mean accuracy at noise standard deviation 1.0 compared to 0.911 for the L2 model, 0.893 for the dual baseline, and 0.842 for the L1 model. Noise resistant models were ranked as follows: L2 > dual > L1. Cross-validation tests were paired. No model superiority was demonstrated, (all p ≥ 0.394). Conclusion. The idempotent decomposition provided a mathematically sound connection from the bicomplex model’s topology, which is useful for component-wise learning. Evidence from the analysis suggests a moderate level of noise resistance is exhibited. More important is the need for wider testing.
Keywords bicomplex numbers; hyperbolic-valued norm; multi-normed spaces; idempotent decomposition; machine learning; regularisation; robust classification.
Published In Volume 14, Issue 3, July-September 2023
Published On 2023-09-08
DOI https://doi.org/10.71097/IJSAT.v14.i3.11478
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Crossref DOI prefix of IJSAT is 10.71097/IJSAT

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