# 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:

$FR(\gamma)_{a,b}:=R(\gamma^{-1}(-\infty,a])_b.$

$$FR(\gamma)$$ 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

$DR(P)_{d,r}:=R(P_{d,r})_r.$

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.

## 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.