🤖 AI Summary
This work addresses the limitation of the Tanimoto kernel (Jaccard index), which is restricted to binary sets or nonnegative real-valued functions. We propose the first generalized Tanimoto kernel for **arbitrary real-valued functions**. Our method decomposes each function into signed magnitude components, mapping it to a pair of signed sets; this yields a rigorously defined set-based representation, from which we derive an explicit feature map and the associated reproducing kernel Hilbert space (RKHS) structure. Building on general kernel design principles, we further provide a piecewise-linear analytic formulation and a differentiable smooth approximation. The resulting framework unifies similarity modeling for real-valued functions, combining theoretical soundness with computational tractability. Empirically, it significantly improves generalization performance in function regression and similarity learning tasks.
📝 Abstract
The Tanimoto kernel (Jaccard index) is a well known tool to describe the similarity between sets of binary attributes. It has been extended to the case when the attributes are nonnegative real values. This paper introduces a more general Tanimoto kernel formulation which allows to measure the similarity of arbitrary real-valued functions. This extension is constructed by unifying the representation of the attributes via properly chosen sets. After deriving the general form of the kernel, explicit feature representation is extracted from the kernel function, and a simply way of including general kernels into the Tanimoto kernel is shown. Finally, the kernel is also expressed as a quotient of piecewise linear functions, and a smooth approximation is provided.
