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

TL;DR

This work generalizes Jaccard/Tanimoto overlap to signed real and complex signals by introducing sign-aware, multistate embeddings that preserve magnitude. It defines a bounded metric $d_{ ext{peak}}$ and a corresponding PSD kernel $K_{ ext{peak}}$ by applying the MinMax/Tanimoto framework to sign-split embeddings, and then extends to multistate partitions $oldsymbol{eta}^{(K)}$ to enable exact Möbius-based coalition budgeting and probabilistic interpretation. A probabilistic view maps signals to measures on a coordinate-state space, linking $d_{ ext{peak}}$ to total variation and enabling a data-processing interpretation under partition coarsening. The framework supports complex-valued signals via Cartesian and polar embeddings, provides scalable ensemble diagnostics through grand-coalition measures, and demonstrates phase-shifted signal behavior in case studies, offering a unified, interpretable tool for correlograms, similarity graphs, and kernel methods with regime–intensity decomposition. Overall, the peak-to-peak distance family fuses metric, kernel, probabilistic, and budget-accounting properties into a single, sign-aware framework with broad applicability to scientific and financial data analysis.

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 Möbius 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.

Figures (1)

• Figure 1: Visualization of the sign-aware peak-to-peak framework. (a) Components of the pairwise metric $d_{\text{peak}}(A,B)$ for Signal A ($\sin(2\pi t/T)$) and Signal B ($\sin(2\pi(t - \tfrac{T}{4})/T)$) with a $90^\circ$ phase shift. The green shaded area represents the sign-aware intersection $N(A,B)$, and the light blue envelope shows the peak-to-peak union $U_{\text{peak}}(A,B)$ used for normalization. (b) Venn diagram illustrating the exclusive intersection components $\tilde{N}(S)$ for an ensemble of three signals with phase shifts $0^\circ$, $30^\circ$, and $90^\circ$, demonstrating the multi-way additive decomposition. Numerical labels are in arbitrary amplitude units with peak amplitude normalized to $1$.

Theorems & Definitions (56)

• Remark 3.1: Zero as a physical state
• Remark 3.2: Semantic separation of channels
• Definition 3.3
• Definition 3.4: Sign-split embedding
• Lemma 3.5: $L^{1}$-preservation of the sign-split embedding
• proof
• Remark 3.6
• Definition 3.7: Index set of sign agreement
• Definition 3.8: Pairwise sign-aware intersection
• Definition 3.9: Peak-to-peak union