Clifford algebra multivectors and kernels for melanoma classification

Abstract

Melanoma is a deadly skin disease. Availability of digital skin lesion datasets ease the exploration of ample classification studies. Both theoretical and heuristics improvements are achieved thanks to these new datasets. Being one of many high‐level feature‐driven classification methods, support vector machines (SVMs) are widely used in the literature as melanoma classifiers. Almost all of these studies are using a limited set of predefined kernels. In this study, we propose a newly developed Clifford kernel for the classification of dermoscopic skin lesions. We develop Clifford‐based linear, polynomial, and exponential kernels in the Clifford algebra (CA) Cℓ5, 0 0‐, 2‐, and 4‐vector subspaces. CAs are noncommutative but associative and distributive over addition. We showed that the newly developed Clifford kernels are embedded into SVM classifiers to successfully identify malignant skin lesions in a binary classification settings. Clifford kernel results are compared with mostly used gaussian and polynomial kernels with real‐valued SVM classifiers. Accuracy of all classifiers are assessed with cross‐validation using imbalanced and balanced datasets of 112, 162, and 192 lesions. SVM kernels in comparison are parameterized to scan wide range of possibilities. We show that Clifford‐based polynomial kernels outperforms in all, balanced and imbalanced, datasets having average accuracy of 83%. The consistence of high accuracies obtained with Clifford polynomial kernel shows that skin lesion features are logically designed and Clifford‐based SVM is able to model class separations in the feature space.

A generalization of Euler's formula in Clifford algebras

“…Akar et al 15 proposed a newly developed Clifford kernel for the classification of dermoscopic skin lesions. We develop Clifford‐based linear, polynomial, and exponential kernels in the Clifford algebra (CA) Cℓ 5, 0 , zero‐, two‐, and four‐vector subspaces.…”