📂Topological Data Analysis

Persistent Modules

Definition ¹

Let $R$ be called a ring and let the extended set of integers $\mathbb{Z}$ be denoted by $\overline{\mathbb{Z}} := \mathbb{Z} \cup \left\{ \pm \infty \right\}$.

1. Given a chain map $f^{i} : \mathsf{C}_{\ast}^{i} \to \mathsf{C}_{\ast}^{i+1}$ between the chain complexes $\mathsf{C}_{\ast}^{i}$ as follows, $\mathcal{C} := \left\{ \left( \mathsf{C}_{\ast}^{i} , \partial_{i} , f^{i} \right) : i \ge 0 \right\}$ is called a Persistent Complex. $$ \mathsf{C}_{\ast}^{0} \overset{f^{0}}{\longrightarrow} \mathsf{C}_{\ast}^{1} \overset{f^{1}}{\longrightarrow} \mathsf{C}_{\ast}^{2} \overset{f^{2}}{\longrightarrow} \cdots $$
2. Given a homomorphism $\varphi^{i} : M^{i} \to M^{i+1}$ between $R$-modules $M^{i}$, $\mathcal{M} := \left\{ \left( M^{i} , \varphi^{i} \right) : i \ge 0 \right\}$ is called a Persistent Module.
3. Each component $\mathsf{C}_{\ast}^{i}$ of a Persistent Complex $\mathcal{C}$ or $M^{i}$ of a Persistent Module $\mathcal{M}$ is a finitely generated $R$-module, and if for some $m \in \mathbb{Z}$ all $f^{i}$ or $\varphi^{i}$ are $i \ge m$ when it is an isomorphism, then $\mathcal{C}$ or $\mathcal{M}$ is referred to as Finite Type.
4. The interval $(i,j)$ that satisfies $0 \le i \le j \in \overline{Z}$ is called a $\mathcal{P}$-interval.

Explanation

What kind of nonsense is this definition?

These definitions are excerpted from the paper by Zomorodian, and as you can see, the three definitions seem unrelated. Reading his paper, the most troublesome part for me was this. Despite the sequential introduction of such definitions, there seems to be no connection among them. The Persistent Complex has nothing to do with being “Persistent”, and the $\mathcal{P}$-interval doesn’t even mention $\mathcal{P}$. Moreover, the Persistent Module is not a ‘module’ but a ‘family of modules’, yet it’s referred to as just a module.

Focusing solely on the meaning of these by themselves will drive one to madness. If someone had just told me ’they’re literally unrelated in reality’, I could have saved more time and studied more comfortably. Let’s take these definitions as they are and move on to the actually important discussion.

Topological Data Analysis

In Topological Data Analysis, the main interest is in how some topological properties persist through the Filtered Complex. For example, if one constructs a Filtered Complex by listing various complexes with increasing radius $\varepsilon > 0$ in Vietoris-Rips Complexes, the change that follows is represented by $f^{i}$. $\partial_{p}$, being the boundary map that must accompany each complex irrespective of the Filtered Complex, is not a concern in this context. This represents the translation of the intuitive process of TDA into mathematical expression, assuming that the Persistent Module satisfies especially manageable conditions through $\mathsf{C}_{\ast}^{i} = M^{i}$ and $f^{i} = \varphi^{i}$. Understanding this context makes the introduction of such definitions no longer seem strange. The problem is that the Persistent Module $\mathcal{M}$, even under such favorable conditions, remains difficult to deal with. Here, the graded module appears like a comet.

When $\mathcal{M}$ is a Persistent Module over $R$, in the context of $R[t]$ $$ \alpha \left( \mathcal{M} \right) := \bigoplus_{i=0}^{\infty} M^{i} $$ you can give a group action $$ t \cdot \left( m^{0} , m^{1} , m^{2} , \cdots \right) = \left( 0, \varphi^{0} \left( m^{0} \right) , \varphi^{1} \left( m^{1} \right) , \varphi^{2} \left( m^{2} \right) , \cdots \right) $$ In simple terms, saying that $t$ acts or multiplies means to promote an element from $M^{i}$ to $M^{i+1}$. This Correspondence $\alpha$ moves the problem of Persistent Modules to that of graded modules according to the following theorem.

Artin-Rees Theory in Commutative Algebra: The correspondence $\alpha$ defines a category equivalence between the category of finite type persistent homology over $R$ and the category of finitely generated standard graded modules over $R[t]$.

Graded modules, while sounding complex, may be nothing more than the polynomial ring we often dealt with in undergraduate or even simpler. If we can accept these buildups, we’re ready to move from the abstract algebraic topology to calculating homology in the realm of algorithms. $$ Q (i,j) := \sum^{i} F[t] / \left( t^{j-i} \right) $$ We will relate the ring, which is also a graded ring and a PID over the field $F$, with the set of $\mathcal{P}$-intervals $\mathcal{S}$ by a surjection $Q$ as follows. Imagine, for example, if our data showed that the $1$th Betti number was $\beta_{1} = 2$ from $\varepsilon_{5} = 0.6$ to $\varepsilon_{12} = 1.5$, we have found a relationship such as $$ Q (5,12) = \sum^{i} F[t] / \left( t^{12-5} \right) $$ and the duration the topological property described by Betti numbers persisted is $1.5 - 0.6 = 0.9$. Although it’s not explicit, listening to this story might give you some empathy as to why the term “Persistent” is used for Persistent Complexes and the like.