Categories and the TDA pipeline

 Conference topic

This post contains topics and ideas from ACAT at HIM, April 2017, as presented by Professor Ulrich Bauer (see slide 11 of his presentation, online at ulrich-bauer.org/persistence-bonn-talk.pdf). The central theme is to assign categories and functors to analyze the process
$$\text{filtration}\ \longrightarrow\ \text{(co)homology}\ \longrightarrow\ \text{barcode.} \hspace{3cm}(\text{pipe})$$ Remark: The categories we will use are below. For filtrations, we have the ordered reals (though any poset $P$ would work) and topological spaces:
$$\begin{align*} R\ :\ & \Obj(R) = \R,  & \Top\ :\ & \Obj(\Top) = \{\text{topological spaces}\}, \\[5pt] & \Hom(r,s) = \begin{cases} \{r \mapsto s\}, & \text{ if } r\leqslant s, \\ \emptyset, & \text{ else,} \end{cases} && \Hom(X,Y) = \{\text{functions }f:X\to Y\}. \end{align*}$$
For (co)homology groups, we have the category of (framed) vector spaces. We write $V^n$ for $V^{\oplus n} = V\oplus V\oplus \cdots \oplus V$, and $e_n$ for a frame of $V^n$ (see below).
$$\begin{align*} \Vect\ :\ & \Obj(\Vect) = \{V^{\oplus n}\ :\ 0\leqslant n< \infty\},\\ & \Hom(V^n,V^m) = \{\text{homomorphisms }f:V^n\to V^m\}, \\[5pt] \Vect^{fr}\ :\ & \Obj(\Vect^{fr}) = \{V^n\times e^n\ :\ 0\leqslant n<\infty\}, \\ & \Hom(V^n\times e^n,V^m\times e^m) = \{\text{hom. }f:V^n\to V^m,\ g:e^n\to e^m,\ g\in \Mat(n,m)\}. \end{align*}$$
Finally for barcodes, we have $\Delta$, the category of finite ordered sets, and its variants. A partial injective function, or matching $f:A\nrightarrow B$ is a bijection $A'\to B'$ for some $A'\subseteq A$, $B'\subseteq B$.
$$\begin{align*} \Delta\ :\ & \Obj(\Delta) = \{[n]=(0,1,\dots,n)\ :\ 0\leqslant n<\infty\},\\ & \Hom([n],[m]) = \{ \text{order-preserving functions }f:[n]\to [m]\}, \\[5pt] \Delta'\ :\ & \Obj(\Delta')= \{a=(a_0<a_1<\cdots<a_n)\ :\ a_i\in \Z_{\geqslant 0}, 0\leqslant n<\infty\},\\ & \Hom(a,b) = \{\text{order-preserving functions }f:a\to b\}, \\[5pt] \Delta''\ :\ & \Obj(\Delta'')= \{a=(a_0<a_1<\cdots<a_n)\ :\ a_i\in \Z_{\geqslant 0}, 0\leqslant n<\infty\},\\ & \Hom(a,b) = \{\text{order-preserving partial injective functions }f:a\nrightarrow b\}. \end{align*}$$

Definition: A frame $e$ of a vector space $V^n$ is equivalently:

• an ordered basis of $V^n$,
• a linear isomorphism $V^n\to V^n$, or
• an element in the fiber of the principal rank $n$ frame bundle over a point.

Frames (of possibly different sizes) are related by full rank elements of $\Mat(n,m)$, which contains all $n\times m$ matrices over a given field.

Definition: Let $(P,\leqslant)$ be a poset. A (indexed topological) filtration is a functor $F:P\to \Top$, with
$$\Hom(F(r),F(s)) = \begin{cases} \{\iota:F(r) \hookrightarrow F(s)\}, & \text{ if }r\leqslant s, \\ \emptyset, & \text{ else,} \end{cases}$$
where $\iota$ is the inclusion map. That is, we require $F(r)\subseteq F(s)$ whenever $r\leqslant s$.

Definition: A persistence module is the composition of functors $M_i:P \tov{F} \Top \tov{H_i} \Vect$.

Homology will be taken over some field $k$. A framed persistence module is the same composition as above, but mapping into $\Vect^{fr}$ instead. The framing is chosen to describe how many different vector spaces have already been encountered in the filtration.

Definition: A barcode is a collection of intervals of $\R$. It may also be viewed as the composition of functors $B_i:P\tov{F}\Top\tov{H_i}\Vect \tov{\dim}\Delta$.

Similarly as above, we may talk about a framed barcode by instead mapping into $\Vect^{fr}$ and then to $\Delta''$, keeping track of which vector spaces we have already encountered. This allows us to interpret the process $(\text{pipe})$ in two different ways. First we have the unframed approach
$$\begin{array}{r c c c l} \Top & \to & \Vect & \to & \Delta, \\ X_t & \mapsto & H_i(X_t;k) & \mapsto & [\dim(H_i(X_t;k))]. \end{array}$$
The problem here is interpreting the inclusion $X_t\hookrightarrow X_{t'}$ as a map in $\Delta$, for instance, in the case when $H_i(X_t;k)\cong H_i(X_{t'};k)$, but $H_i(X_t\hookrightarrow X_{t'}) \neq \id$. To fix this, we have the framed interpretation of $(\text{pipe})$
$$\begin{array}{r c c c l} \Top & \to & \Vect^{fr} & \to & \Delta'', \\ X_t & \mapsto & H_i(X_t;k)\times e & \mapsto & [e]. \end{array}$$
The first map produces a frame $e$ of size $n$, where $n$ is the total number of different vector spaces encountered over all $t'\leqslant t$, by setting the first $\dim(H_i(X_t;k))$ coordinates to be the appropriate ones, and then the rest. This is done with the second map to $\Delta''$ in mind, as the size of $[e]$ is $\dim(H_i(X_t;k))$, with only the first $\dim(H_i(X_t;k))$ basis vectors taken from $e$. As usual, these maps are best understood by example.

Example: Given the closed curve $X$ in $\R^2$ below, let $\varphi:X\to \R$ be the height map from the line 0, with $X_i=\varphi^{-1}(-\infty,i]$, for $i=r,s,t,u,v$. Let $e_i$ be the standard $i$th basis vector in $\R^N$.

Remark: This seems to make $(\text{pipe})$ functorial, as the maps $X_t\hookrightarrow X_{t'}$ may be naturally viewed as partial injective functions in $\Delta''$, to account for the problem mentioned with the unframed interpretation. However, we have traded locality for functoriality, as the image of $X_t$ in $\Delta''$ can not be calculated without having calculated $X_{t'}$ for all $t'<t$.

References: Bauer (Algebraic perspectives of persistence), Bauer and Lesnick (Induced matchings and the algebraic stability of persistence barcodes)