Return to Question

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as \begin{equation} J(a,b) = 1 - \frac{||a\odot b||_1}{||a\odot b||_1+||a-b||_1} \end{equation} where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.

I've tried with the naive approach. After some messy algebra, I need to prove the following \begin{equation} ||a-b||_1||a\odot c||_1||b\odot c||_1 + ||a-b||_1||b-c||_1||a\odot c||_1 + ||a-b||_1 ||b-c||_1 ||a-c||_1 + ||b-c||_1||a\odot b||_1 ||a\odot c||_1 + ||a-b||_1||b-c||_1||a\odot c||_1 \geq ||a-c||_1||a\odot b||_1 ||b\odot c||_1. \end{equation}

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as \begin{equation} J(a,b) = 1 - \frac{\|a\odot b\|_1}{\|a\odot b\|_1+\|a-b\|_1} \end{equation} where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.

I've tried with the naive approach. After some messy algebra, I need to prove the following \begin{equation} \|a-b\|_1\|a\odot c\|_1\|b\odot c\|_1 + \|a-b\|_1\|b-c\|_1\|a\odot c\|_1 + \|a-b\|_1 \|b-c\|_1 \|a-c\|_1 + \|b-c\|_1\|a\odot b\|_1 \|a\odot c\|_1 + \|a-b\|_1\|b-c\|_1\|a\odot c\|_1 \geq \|a-c\|_1\|a\odot b\|_1 \|b\odot c\|_1. \end{equation}

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as

 \begin{equation} J(a,b) = 1 - \frac{||a\odot b||_1}{||a\odot b||_1+||a-b||_1} \end{equation} where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as  \begin{equation} J(a,b) = 1 - \frac{||a\odot b||_1}{||a\odot b||_1+||a-b||_1} \end{equation} where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.

I've tried with the naive approach. After some messy algebra, I need to prove the following \begin{equation} ||a-b||_1||a\odot c||_1||b\odot c||_1 + ||a-b||_1||b-c||_1||a\odot c||_1 + ||a-b||_1 ||b-c||_1 ||a-c||_1 + ||b-c||_1||a\odot b||_1 ||a\odot c||_1 + ||a-b||_1||b-c||_1||a\odot c||_1 \geq ||a-c||_1||a\odot b||_1 ||b\odot c||_1. \end{equation}

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as

\begin{equation} J(a,b) = 1 - \frac{||ab||_1}{||ab||_1+||a-b||_1}. \end{equation}

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.

Define the Jaccard distance between two continuous vectors $a, b\in [0,1]^p$ as

\begin{equation} J(a,b) = 1 - \frac{||a\odot b||_1}{||a\odot b||_1+||a-b||_1} \end{equation} where $\odot$ is the Hadamard product (element-wise product).

Is it a metric? Note that $a,b \in [0,1]^p$ rather than $\{0,1\}^p$.