Functorial persistence

The goal of this post is to overcome some hurdles encountered by Bauer and Lesnick. In their approach, some geometric information is lost in passing from persistence modules to matchings. Namely, if an interval ends, we forget if the  $k$-cycle it represents becomes part of another $k$-cycle or goes to 0. Recall:

• $(\R,\leqslant)$ is the category of real numbers and unique morphisms $s\to t$ whenever $s\leqslant t$,
• $\Vect$ ($\BVect$) is the category of (based) finite dimensional vector spaces, and
• $\Set_*$ is the category of pointed sets.

We begin by recalling all the classical notions in the TDA pipeline.

Defintion: A persistence module is a functor $F:(\R,\leqslant)\to \Vect$. The barcode of a persistence module $F$ is a collection of pairs $(I,k)$, where $I\subseteq \R$ is an interval and $k\in \Z_{>0}$ is a positive integer.

Crawley-Boevey describes how to find the decomposition of a persistence module into interval modules. The $k$ for each $I$ is usually 1, but is 2 (and more) if the same interval appears twice (or more) in the decomposition. A barcode contains the same information as a \emph{persistence diagram}, though the former is drawn as horizontal bars and the latter is presented on a pair of axes.

Definition: A matching $\chi$ of barcodes $\{(I_i,k_i)\}_i$ and $\{(J_j,\ell_j)\}_j$ is a bijection $I'\to J'$, for some $I'\subseteq \{(I_i,k_i)\}_i$ and $J'\subseteq \{(J_j,\ell_j)\}_j$.

We write matchings as $\chi\colon \{(I_i,k_i)\}_i \nrightarrow \{(J_j,\ell_j)\}_j$.

Definition: A filtered persistence module is a functor $F:(\R,\leqslant) \to \BVect$ for which $F(s\leqslant t)(e_i) =f_j$ or 0, for every $e_i$ in the basis of $F(s)$ and $f_j$ in the basis of $F(t)$.

The notion of filtered persistence module is used for a stronger geometric connection. Indeed, for every filtered space $X$ the persistence module along this filtration is also filtered (once interval modules have been found), as then inclusions $X_s\hookrightarrow X_t$ will induce isomorphisms in homology onto their image. That is, a pair of homology classes from the source may combine in the target, but if the classes come from interval modules, a class from the source can not be in two non-homologous classes of the target.

Remark: The above dicussion highlights that choosing a basis in the definition of a persistence module already uses the decomposition of persistence modules into interval modules.

It is immediate that a morphism of persistence modules is a natural transformation. Let $\BPVect$ be the full subcategory of $\BVect$ consisting of elements in the image of some filtered persistence module (the objects are the same, we just have a restriction of allowed morphisms).

Definition:  Let $\mathcal B$ be the functor defined by \[ \begin{array}{r c l}
\mathcal B\colon \BPVect & \to & \Set_*, \\
(V,\{e_1,\dots,e_n\}) & \mapsto & \{0,1,\dots,n\}, \\
\left(\varphi:(V,\{e_i\}) \to (W,\{f_j\})\right) & \mapsto & \left(
i \mapsto \begin{cases}
j & \text{ if } \varphi(e_i) = f_j, \\ 0 & \text{ if } \varphi(e_i)=0 \text{ or } i=0.
\end{cases} \right)
\end{array} \]

The basepoint of every set in the image of $\mathcal B$ is 0.

Definition: Let $F,G$ be persistence modules and $\eta$ a morphism $F\to G$.

• The persistence diagram of $F$ is the functor $\mathcal B\circ F$.
• The matching induced by $\eta$ is the natural transformation $\mathcal B(\eta): \mathcal B\circ F\to \mathcal B\circ G$.

Bauer and Lesnick's definition of "matching" allow for more freedom to mix and match barcode intervals, but this also restricts how much information of a persistence module morphism can be tracked.

Example: The following example has a horizontal filtration with the degree 0 homology barcode on the left and the degree 1 homology barcode on the right. Linear maps of based vector spaces have also been shown to indicate how homology classes are born, die (column of zeros), and combine (row with more than one 1).

Example: Bauer and Lesnick present Example 5.6 to show that functoriality does not work in their setting. We reproduce their example and show that functoriality does work in our setting. Note that vertical ordering of the bars does not matter once they are named.

Apply the functor $\mathcal B$ to the whole diagram to get the matchings induced by $\eta$ and $\xi$, as below.

Next we hope to understand how interleavings fit into this setup.

References: Bauer and Lesnick (Induced matchings and the algebraic stability of persistence barcodes), Crawley-Boevey (Decomposition of pointwise finite-dimensional persistence modules)