3. 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 \(a_2 \leq b_2\).

3.1. 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.

3.2. Function-Rips Bifiltration

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:


\(FR(\gamma)\) is always 1-critical.

We mention three natural choices of \(\gamma\), each of which is implemented in RIVET:

  • A ball density function, defined by
\[\gamma(x)=C\cdot (\# \text{ points in } P \text{ within distance }r \text{ of }x),\]

where \(r>0\) is a fixed parameter, the “radius”, and \(C\) is a normalization constant, chosen so that \(\sum_{x\in P} \gamma(x)=1\).

  • A Gaussian density function, given by
\[\gamma(x)=C\sum_{y\in P} e^{\frac{-d(x,y)^2}{2\sigma}},\]

where \(\sigma>0\) is a parameter, the “standard deviation,” and \(C\) is a normalization constant.

  • An eccentricity function, i.e.,
\[\gamma(x):= \left(\frac{\sum_{y\in P} d(x,y)^q}{|P|}\right)^{\frac{1}{q}},\]

where \(q\in [1,\infty)\) is a parameter.

3.3. Degree-Rips Bifiltration

For \(r,d\in \mathbb R\), let \(P_{d,r}\subset P\) be the set of vertices in \(N(P)_r\) of degree at least \(d\). We define the degree-Rips bifiltration \(DR(P)\) by taking


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.

3.4. 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.

3.5. 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 \(\displaystyle\oplus_{c\in \mathcal B}\ \mathcal I^c\) 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.

3.6. 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 there exists a presentation \(f:F\to G\) for \(M\) 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.

3.7. 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.

3.8. 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.

3.9. 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 \(F^i\).

3.10. 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 \(M\). The coarsening operation depends on a choice of finite grid \(G\subset\mathbb R^2\), such that \(G\) contains some upper bound of 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.