Interleaving distance

A persistence module ${\displaystyle \mathbb {V} }$ is a collection ${\displaystyle (V_{t}\mid t\in \mathbb {R} )}$ of vector spaces indexed over the real line, along with a collection ${\displaystyle (v_{t}^{s}:V_{s}\to V_{t}\mid s\leq t)}$ of linear maps such that ${\displaystyle v_{t}^{t}}$ is always an isomorphism, and the relation ${\displaystyle v_{t}^{s}\circ v_{s}^{r}=v_{t}^{r}}$ is satisfied for every ${\displaystyle r\leq s\leq t}$. The case of ${\displaystyle \mathbb {R} }$ indexing is presented here for simplicity, though the interleaving distance can be readily adapted to more general settings, including multi-dimensional persistence modules.[7]

Let ${\displaystyle \mathbb {U} }$ and ${\displaystyle \mathbb {V} }$ be persistence modules. Then for any ${\displaystyle \delta \in \mathbb {R} }$, a ${\displaystyle \delta }$-shift is a collection ${\displaystyle (\phi _{t}:U_{t}\to V_{t+\delta }\mid t\in \mathbb {R} )}$ of linear maps between the persistence modules that commute with the internal maps of ${\displaystyle \mathbb {U} }$ and ${\displaystyle \mathbb {V} }$.

The persistence modules ${\displaystyle \mathbb {U} }$ and ${\displaystyle \mathbb {V} }$ are said to be ${\displaystyle \delta }$-interleaved if there are ${\displaystyle \delta }$-shifts ${\displaystyle \phi _{t}:U_{t}\to V_{t+\delta }}$ and ${\displaystyle \psi _{t}:V_{t}\to U_{t+\delta }}$ such that the following diagrams commute for all ${\displaystyle s\leq t}$.

It follows from the definition that if ${\displaystyle \mathbb {U} }$ and ${\displaystyle \mathbb {V} }$ are ${\displaystyle \delta }$-interleaved for some ${\displaystyle \delta }$, then they are also ${\displaystyle (\delta +\varepsilon )}$-interleaved for any positive ${\displaystyle \varepsilon }$. Therefore, in order to find the closest interleaving between the two modules, we must take the infimum across all possible interleavings.

The interleaving distance between two persistence modules ${\displaystyle \mathbb {U} }$ and ${\displaystyle \mathbb {V} }$ is defined as ${\displaystyle d_{I}(\mathbb {U} ,\mathbb {V} )=\inf\{\delta \mid \mathbb {U} {\text{ and }}\mathbb {V} {\text{ are }}\delta {\text{-interleaved}}\}}$.[8]