Bandwidth adjustable Tanimoto kernel: a smooth alternative to the Gaussian kernel

Abstract Kernel methods have long played a central role in machine learning by enabling nonlinear modeling in high dimensional feature spaces. Among them, the Gaussian Radial Basis Function (RBF) kernel is widely used due to its smoothness, universality, and excellent performance in supervised learning. In contrast, the Jaccard/Tanimoto coefficient, originally defined over sets and binary vectors, has achieved notable success in chemoinformatics and information retrieval. Recent studies have shown that the Jaccard/Tanimoto coefficient can be extended to real-valued vectors and interpreted as a valid kernel. In this paper, we propose a natural generalization of the Tanimoto kernel by introducing a bandwidth-like parameter, leading to the Bandwidth-Adjustable Tanimoto (BAT) kernel. We focus on the shared structure of the BAT and the Gaussian RBF kernels in terms of their infinite polynomial expansions and investigate the universality of the BAT kernels, clarifying their relationship to the universal Taylor kernels. In particular, we show that by incorporating a bias term, the BAT kernel attains the universal approximation property comparable to that of the Gaussian kernel. Through experiments on classification, regression, and clustering tasks using benchmark datasets, we demonstrate that the BAT kernel performs comparably to the Gaussian kernel, while even the original Tanimoto kernel achieves competitive results. In particular, the BAT kernel attains performance on par with the Gaussian kernel in classification tasks, and it slightly surpasses the Gaussian kernel in regression and clustering tasks. A comparison of their functional forms, together with experiments on both regression and classification tasks involving nonlinear structures, further shows that the BAT kernel yields smoother predictive surfaces than the RBF kernel. Overall, our results indicate that the BAT kernel offers universality and practical performance, making it an alternative to the Gaussian kernel.