Interpolating between the Jaccard distance and an analogue of the normalized information distance

TL;DR

This paper explores a family of metrics interpolating between the Jaccard distance and an analogue of normalized information distance, providing formal verification and characterizing conditions for metric properties.

Contribution

It characterizes when the symmetric Tversky ratio models form metrics and introduces new interpolating metrics, verified in the Lean proof assistant.

Findings

D_{α,β} is a metric iff 0≤α≤1/2 and β≥1/(1−α)

Extreme points include the Jaccard distance and an analogue of normalized information distance

Introduces a family of metrics V_p interpolating between these distances

Abstract

Jim\'enez, Becerra, and Gelbukh (2013) defined a family of "symmetric Tversky ratio models" $S_{\alpha,\beta}$, $0\le\alpha\le 1$, $\beta>0$. Each function $D_{\alpha,\beta}=1-S_{\alpha,\beta}$ is a semimetric on the powerset of a given finite set. We show that $D_{\alpha,\beta}$ is a metric if and only if $0\le\alpha \le \frac12$ and $\beta\ge 1/(1-\alpha)$. This result is formally verified in the Lean proof assistant. The extreme points of this parametrized space of metrics are $\mathcal V_1=D_{1/2,2}$, the Jaccard distance, and $\mathcal V_{\infty}=D_{0,1}$, an analogue of the normalized information distance of M. Li, Chen, X. Li, Ma, and Vit\'anyi (2004). As a second interpolation, in general we also show that $\mathcal V_p$ is a metric, $1\le p\le\infty$, where $$\Delta_p(A,B)=(|B\setminus A|^p+|A\setminus B|^p)^{1/p},$$ $$\mathcal V_p(A,B)=\frac{\Delta_p(A,B)}{|A\cap B| + \Delta_p(A,B)}.$$