Sign-Aware Multistate Jaccard Kernels and Geometry for Real and Complex-Valued Signals

TL;DR

This paper introduces a novel sign-aware multistate Jaccard framework for measuring similarity in real and complex signals, providing bounded metrics, kernels, and probabilistic interpretations for advanced signal analysis.

Contribution

It extends overlap-based distances to arbitrary signals with a measure-theoretic geometry, enabling new similarity measures and kernel methods for complex-valued signals.

Findings

Defines a set- and measure-theoretic geometry for signals

Develops a family of positive-semidefinite kernels and distances

Provides interpretable, probabilistic, and regime-aware signal similarity measures

Abstract

We introduce a sign-aware, multistate Jaccard/Tanimoto framework that extends overlap-based distances from nonnegative vectors and measures to arbitrary real- and complex-valued signals while retaining bounded metric and positive-semidefinite kernel structure. Formally, the construction is a set- and measure-theoretic geometry: signals are represented as atomic measures on a signed state space, and similarity is given by a generalized Jaccard overlap of these measures. Each signal is embedded into a nonnegative multistate representation, using positive/negative splits for real signals, Cartesian and polar decompositions for complex signals, and user-defined state partitions for refined regime analysis. Applying the Tanimoto construction to these embeddings yields a family of $[0,1]$ distances that satisfy the triangle inequality and define positive-semidefinite kernels usable directly in kernel methods and graph-based learning. Beyond pairwise distances, we develop coalition analysis via M\"obius inversion, which decomposes signal magnitude into nonnegative, additive contributions with exact budget closure across coalitions of signals. Normalizing the same embeddings produces probability measures on coordinate -- state configurations, so that the distance becomes a monotone transform of total variation and admits a regime -- intensity decomposition. The resulting construction yields a single, mechanistically interpretable distance that simultaneously provides bounded metric structure, positive-semidefinite kernels, probabilistic semantics, and transparent budget accounting within one sign-aware framework, supporting correlograms, feature engineering, similarity graphs, and other analytical tools in scientific and financial applications.