# Mathematical Preliminaries¶

To prepare for a detailed explaination of what RIVET can do and how it is used, we review some basic mathematical notions and establish some terminology.

To start, we define a partial order on \(\mathbb R^2\) by taking \(a \leq b\) if and only if \(a_1 \leq b_1\) and \(a_2 \leq b_2\).

## Bifiltrations¶

A *bifiltration* \(F\) is a collection of finite simplicial complexes indexed by \(\mathbb R^2\) such that \(F_a\subset F_b\) whenever \(a\leq b\). In the computational setting, the bifiltrations \(F\) we encounter are always *essentially finite*. This finiteness condition can be specified succinctly in the language of category theory: \(F\) is essentially finite if \(F\) is a left Kan extension of a diagram indexed by a finite grid in \(\mathbb R^2\). (See the RIVET paper for a more elementary definition.) Such \(F\) can be specified by a single simplicial complex \(S\) (the colimit of \(F\) ) together with a collection of incomparable points \(\mathrm{births}(\sigma)\subset\mathbb R^2\) for each simplex \(\sigma\in S\), specifying the bigrades at which \(\sigma\) is born. If \(\mathrm{births}(\sigma)\) contains one element for each \(\sigma\in S\), then we say \(F\) is *one-critical*. Otherwise, we say \(F\) is *multi-critical*.

We next introduce two contructions of bifiltrations from data.

## Function-Rips Bifiltraiton¶

For \(P\) a finite metric space and \(r\geq 0\), let \(N(P)_r\) denote the \(r\)-neighborhood graph of \(P\), i.e., the vertex set of \(N(P)_r\) is \(P\), and edge \([i,j]\in N(P)_r\) if and only if \(d(i,j)\leq r\). If \(r<0\), we define \(N(P):=\emptyset.\) We define the *Vietoris-Rips complex* \(R(P)_r\) to be the clique complex on \(N(P)_r\), i.e. the largest simplical complex with 1-skeleton \(N(P)_r\).

Given a finite metric space \(P\) and any function \(\gamma:P\to \mathbb R\), we define the *function-Rips* bifiltration \(FR(\gamma)\) as follows: \[FR(\gamma)_{a,b}:=R(\gamma^{-1}(-\infty,a])_b.\] \(FR(\gamma)\) is always 1-critical.

\(\gamma\) is often chosen to be a density estimate on \(P\). Another common choice is to take \(\gamma\) to be a coeccentricity function on \(P\), e.g., \(\gamma(x):= \sum_{y\in P} d(x,y)\).

## Degree-Rips Bifiltration¶

For \(d\in \mathbb R\), let \(N(P)_{d,r}\) be the subgraph of \(N(P)_r\) obtained by removing all vertices of degree less than \(d\). We define the *degree-Rips bifiltration* \(DR(P)\) by taking \(DR(P)_{d,r}:=N(P)_{d,r}.\) (Note that this is in fact a bifiltration indexed by \(\mathbb R^{\mathrm{op}}\times \mathbb R\), where \(\mathbb R^{\mathrm{op}}\) denotes the opposite poset of \(\mathbb R\); that is, \(DR(P)_{a,b}\subset DR(P)_{a’,b’}\) whenever \(a\geq a’\) and \(b\leq b’\).). If \(P\) has more than one point, then \(DR(P)\) is multi-critical.

## Bipersistence Modules¶

Let us fix a field \(K\). A *bipersistence module* (also called a 2-D persistence module or 2-parameter persistence module in the literature) \(M\) is a diagram of \(K\)-vector spaces indexed by \(\mathbb R^2\). That is, \(M\) is a collection of vector spaces \(\{M_a\}_{a\in \mathbb{R^2}}\), together with a collection of linear maps \[\{M_{a,b}:M_a\to M_b\}_{a\leq b}\] such that \(M_{a,a}=\mathrm{Id}_{M_a}\) and \(M_{b,c}\circ M_{a,b}=M_{a,c}\) for all \(a \leq b\leq c\).

A *morphism* \(f:M\to N\) of bipersistence modules is a collection of maps \[\{f_a:M_a\to N_a\}_{a\in \mathbb R^2}\]such that \[f_b\circ M_{a,b}= N_{a,b} \circ f_a\] for all \(a\leq b\in \mathbb R^2\). This definition of morphism gives the bipersistence modules the structure of an abelian category; thanks in part to this, many usual constructions for modules from abstract algebra have analogues for bipersistence modules. In particular, direct sums and quotients are well defined.

## Free Persistence Modules¶

For \(c \in \mathbb R^2\), define the bipersistence module \(\mathcal I^c\) by \[\mathcal I^c_a= \begin{cases} K &\mathrm{if }\ a\geq c,\ 0 & \mathrm{otherwise.} \end{cases} \qquad \mathcal I^c_{a,b}= \begin{cases} \mathrm{Id}_K &\mathrm{if }\ a\geq c,\ 0 & \mathrm{otherwise.} \end{cases} \] Note that the support of \(\mathcal I^a\) is the closed upper quadrant in \(\mathbb R^2\) with lower left corner at \(a\).

A *free bipersistence module* is one isomorphic to \[\oplus_{c\in \mathcal B}\ \mathcal I^c\] for some multiset \(\mathcal B\) of points in \(\mathbb R^2\).
There is a natural definition of basis for free modules, generalizing the definition of bases for vector spaces in linear algebra. In close analogy with linear algebra, a morphism \(f:M\to N\) of finitely generated free modules can be represented by a matrix, with respect to a choice of ordered bases for \(M\) and \(N\). Thus, to encode the isomorphism type of \(f\), it enough to store a matrix, together with a bigrade label for each row and each column of the matrix; the labels specify \(M\) and \(N\) up to isomorphism.

## Presentations¶

A *presentation* of a bipersistence module \(M\) is a map \(f:F\to G\) such that \(M\cong G/\mathrm{im}\ f\). We say \(M\) is finitely presented if \(F\) and \(G\) can be chosen to be finitely generated. If \(M\) is finitely presented then, up to isomorphism, there exists a presentation \(f:F\to G\) such that both \(F\) and \(G\) are minimial, i.e., for any other presentation \(f’:F’\to G’\), \(F\) is a summand of \(F’\) and \(G\) is a summand of \(G’\). We call such a presentation *minimal*. Minimal presentations are unique up to isomorphism, but importantly, their matrix representations are non-unique.

## FIReps (Short Chain Complexes of Free Modules)¶

We define a *FIRep* to be chain complex of free bipersistence modules of length 3. Explicitly, then, an firep is a sequence of free bipersistence modules
\[ C_2 \xrightarrow{f} C_1 \xrightarrow{g} C_0. \]
such that \(g\circ f=0\). Associated to an firep is a unique homology module \(\ker g/\mathrm{im}\ f\). A presentation of a bipersistence module can be thought of as a special case of an FIRep, where the last module is trivial.

## Homology of a Bifiltration¶

Applying \(i^{\mathrm{th}}\) simplicial homology with coefficients in \(K\) to each simplicial complex and each inclusion map in a bifiltration \(F\) yields a bipersistence module \(H_i(F)\). If \(F\) is essentially finite, then \(H_i(F)\) is finitely presented.

\(H_i(F)\) is in fact the \(i^{\mathrm{th}}\) homology module of a chain complex \( C(F)\) of bipersistence modules whose value at each point in \(a\in \mathbb R^2\) is the simplical chain complex of \(F_a\). If \(F\) is one-critical, each module of \(C(F)\) is free. In general, \(C(F)\) needn’t be free, but given the portion of \(C(F)\) at indexes \(i-1,\) \(i\), and \(i+1\), it is easy to construct an FIRep whose homology is \(H_i(F)\); this is an observation of Chacholski et al.

## Invariants of a Bipersistence Module¶

As mentioned above, RIVET computes and visualizes three simple invariants of a bipersistence module \(M\):

- The
*fibered barcode*, i.e., the function sending each affine line \(L\subset \mathbb R^2\) with non-negative slope to the barcode \(\mathcal B(M^L)\), where \(M^L\) denotes the restriction of \(M\) along \(L\). - The
*Hilbert function*, i.e., the function \(\mathbb R^2\to \mathbb N\) which sends \(a\) to \(\dim M_a\). - The
*bigraded Betti numbers*\(\xi_i^M\). These are functions \(\mathbb{R}^2 \to \mathbb{N}\) that, respectively, count the number of births, deaths, and “relations amongst deaths” at each bigrade. Formally, given \(r \in \mathbb{R}^2\) and a minimal free resolution $$0 \to F^2\to F^1\to F^0$$ for \(M\), \(\xi_i^M(r)\) is the number of elements at bigrade \(r\) in a basis for \(F^i\).

## Coarsening a Persistence Module¶

Given a finitely presented bipersistence module \(M\), we can *coarsen* \(M\) to obtain an algebraically simpler module carrying approximately the same persistence information as \(M\). As we will describe it here, the coarsening operation depends on a choice of finite grid \(G\subset\mathbb R^2\), such that \(G\) contains some element ordered after all points in the support of the Betti numbers of \(M\). The coarsened module, denoted \(M^G\), is defined by taking \(M^G_a:= M_g\), where \(g\in G\) is the minimum grid element such that \(a\leq g\). The internal maps of \(M^G\) are induced by those of \(M\) in the obvious way.