Here we revisit and expand on persistent homology, previously in the post "Persistent homology (an example)," 2016-05-19. All homology, except where noted, will be over a field $k$, and $X$ will be a topological space. Often a Morse-type function $f:X\to \R$ is introduced along with $X$, but we will try to take a more abstract view.
Definition: The space $X$ may be described as a filtered space with a filtration of sublevel sets
\[ \emptyset = X_0 \subseteq X_1\subseteq \cdots \subseteq X_m = X, \] whose persistence module is the (not necessarily exact) sequence
\[ 0 = H(X_0) \to H(X_1)\to\cdots \to H(X_m) = H(X) \] of homology groups of the filtration.
Remark: Every persistence module may be uniquely decomposed as a direct sum of sequences $0\to k\to \cdots\to k\to 0$, where every map is $\id$, except the first and last. The indices at which each sequence in the summand has its first and last non-zero map are called the birth and death of the homology class represented by the sequence.
In some cases a homology class may not die, so we consider the extended persistence module to make everything finite. We introduce the superlevel sets $X^i = X\setminus X_i$. If $f$ was our Morse-type function for $X$, with critical points $p_1<\cdots<p_m$, then for $t_0<p_1<t_1<\cdots<p_m<t_m$, we set $X_i = f^{-1}(-\infty,t_i]$ and $X^i = f^{-1}[t_i,\infty)$. The extended persistence module of $X$ is
\[
0 = H_k(X_0) \to H_k(X_1)\to\cdots \to H_k(X_m) \to H_k(X,X^m) \to H_k(X,X^{m-1}) \to \cdots \to H_k(X,X^0)=0.
\]
Definition: The persistence of a homology class in a persistence module conveys the idea of how long it is alive, presented by a persistence pair.
The persistence of all homology classes in a persistence module is often presented in a persistence diagram, the collection of persistence pairs $(i,j)$, or $(p_i,p_j)$ or $(f(p_i),f(p_j))$, as desired; or a linear barcode, the collection of persistence pairs $(i,j)$ as intervals $[i,j]$, ordered vertically.
Example: Let $X = T^n=(S^1)^n$ be the $n$-torus. One filtration of $X$ is $X_0=\emptyset$ and $X_i = T^i$ for $1\leqslant i\leqslant n$. Note that $H_k(T^n,T^n\setminus X_n)=H_k(T^n)$ and $H_k(T^n,T^n\setminus X_0)=H_k(\emptyset)$. The first $n+1$ modules of the extended persistence module at level $k$ split into $\binom nk$ sequences, as $H_k(T^n) = \Z^{\binom nk}$. Geometric considerations allow $X^i = T^n\setminus T^i$ to be simplified in some cases. For instance, when $n=3$ and $k=0,1$ we have that $\widetilde H_k(T^3,T^3\setminus T^2)\cong \widetilde H_k(T^3,T^2) \cong \widetilde H_k(T^3/T^2)$, and knowing that $X^1=T^3\setminus T^1\simeq (S^1\vee S^1)\times S^1$, the relevant part of the long exact sequence for relative homology is
The two 1-cycles from $S^1\vee S^1\subset X^1$ map via $f$ to the same 1-cycle in $T^3$, hence $\text{im}(g)=\Z^2$. By exactness, $\text{ker}(g)=\Z^2$, and as $g$ is surjective, $A=\Z$. Hence the extended persistence $k$-modules decompose as
The persistence pairs are $(1,3)$ with multiplicity 2 and $(2,3)$, $(3,1)$ with multiplicity 1. The persistence diagrams and barcodes of the degree 0 and 1 homology classes are given below.
The diagonal $y=x$ is often given to indicate how short a lifespan a class has. Barcodes are usually not given for extended persistence diagrams, as length of a class (birth to death) is less important than position (above or below the diagonal).
Now we consider some generalizations of the ideas presented above.
Remark: A filtration can also be viewed as a diagram $X_0 \to X_1 \to \cdots \to X_m$, where each arrow is the inclusion map. We could generalize and consider a zigzag diagram, a sequence $X_0 \leftrightarrow X_1 \leftrightarrow \cdots \leftrightarrow X_m$, where $\leftrightarrow$ represents either $\to$ or $\leftarrow$. Homology can be applied and the resulting seuquence can also be uniquely decomposed into summands $k \leftrightarrow \cdots \leftrightarrow k$ where every arrow is the identity, giving zigzag persistent homology.
Remark: A filtration could also be viewed as a functor $F:\{0,\dots,m\}\to \text{Top}$, where $F(i)=X_i$ and $F(i\to j)$, for $j\>i$, is the composition of maps $X_i\to \cdots \to X_j$. Hence the degree-$k$ persistent homology of $X_i$ can be defined as the image of the maps $H_kF(i\to j)$, for all $j\>i$, and the functor $H_kF:\{0,\dots,m\}\to \text{Vec}$ may be viewed as the $k$th persistence module. This is a categorification of persistent homology.
Remark: A space $X$ can be filtered in several different ways. A multifiltration $X_\alpha$, for $\alpha$ a multi-index, is a collection of filtrations such that fixing all but one of the indices in $\alpha$ gives a (one-dimensional) filtration of $X$. The multidimensional persistence of $X_\alpha$ is a $|\alpha|$-dimensional grid of homology groups, with the barcode generalizing to the rank invariant, a map on the grid.
Another generalization, viewing filtrations as quivers, will not be discussed here, but rather presented as a separate post later.
References: Edelsbrunner and Morozov (Persistent homology: theory and practice), Carlsson, de Silva, and Morozov (Zigzag persistent homology and real-valued functions), Bubenik and Scott (Categorification of persistent homology), Carlsson and Zomorodian (The theory of multidimensional persistence)
Definition: The space $X$ may be described as a filtered space with a filtration of sublevel sets
\[ \emptyset = X_0 \subseteq X_1\subseteq \cdots \subseteq X_m = X, \] whose persistence module is the (not necessarily exact) sequence
\[ 0 = H(X_0) \to H(X_1)\to\cdots \to H(X_m) = H(X) \] of homology groups of the filtration.
Remark: Every persistence module may be uniquely decomposed as a direct sum of sequences $0\to k\to \cdots\to k\to 0$, where every map is $\id$, except the first and last. The indices at which each sequence in the summand has its first and last non-zero map are called the birth and death of the homology class represented by the sequence.
In some cases a homology class may not die, so we consider the extended persistence module to make everything finite. We introduce the superlevel sets $X^i = X\setminus X_i$. If $f$ was our Morse-type function for $X$, with critical points $p_1<\cdots<p_m$, then for $t_0<p_1<t_1<\cdots<p_m<t_m$, we set $X_i = f^{-1}(-\infty,t_i]$ and $X^i = f^{-1}[t_i,\infty)$. The extended persistence module of $X$ is
\[
0 = H_k(X_0) \to H_k(X_1)\to\cdots \to H_k(X_m) \to H_k(X,X^m) \to H_k(X,X^{m-1}) \to \cdots \to H_k(X,X^0)=0.
\]
Definition: The persistence of a homology class in a persistence module conveys the idea of how long it is alive, presented by a persistence pair.
Example: Let $X = T^n=(S^1)^n$ be the $n$-torus. One filtration of $X$ is $X_0=\emptyset$ and $X_i = T^i$ for $1\leqslant i\leqslant n$. Note that $H_k(T^n,T^n\setminus X_n)=H_k(T^n)$ and $H_k(T^n,T^n\setminus X_0)=H_k(\emptyset)$. The first $n+1$ modules of the extended persistence module at level $k$ split into $\binom nk$ sequences, as $H_k(T^n) = \Z^{\binom nk}$. Geometric considerations allow $X^i = T^n\setminus T^i$ to be simplified in some cases. For instance, when $n=3$ and $k=0,1$ we have that $\widetilde H_k(T^3,T^3\setminus T^2)\cong \widetilde H_k(T^3,T^2) \cong \widetilde H_k(T^3/T^2)$, and knowing that $X^1=T^3\setminus T^1\simeq (S^1\vee S^1)\times S^1$, the relevant part of the long exact sequence for relative homology is
The two 1-cycles from $S^1\vee S^1\subset X^1$ map via $f$ to the same 1-cycle in $T^3$, hence $\text{im}(g)=\Z^2$. By exactness, $\text{ker}(g)=\Z^2$, and as $g$ is surjective, $A=\Z$. Hence the extended persistence $k$-modules decompose as
Now we consider some generalizations of the ideas presented above.
Remark: A filtration can also be viewed as a diagram $X_0 \to X_1 \to \cdots \to X_m$, where each arrow is the inclusion map. We could generalize and consider a zigzag diagram, a sequence $X_0 \leftrightarrow X_1 \leftrightarrow \cdots \leftrightarrow X_m$, where $\leftrightarrow$ represents either $\to$ or $\leftarrow$. Homology can be applied and the resulting seuquence can also be uniquely decomposed into summands $k \leftrightarrow \cdots \leftrightarrow k$ where every arrow is the identity, giving zigzag persistent homology.
Remark: A filtration could also be viewed as a functor $F:\{0,\dots,m\}\to \text{Top}$, where $F(i)=X_i$ and $F(i\to j)$, for $j\>i$, is the composition of maps $X_i\to \cdots \to X_j$. Hence the degree-$k$ persistent homology of $X_i$ can be defined as the image of the maps $H_kF(i\to j)$, for all $j\>i$, and the functor $H_kF:\{0,\dots,m\}\to \text{Vec}$ may be viewed as the $k$th persistence module. This is a categorification of persistent homology.
Remark: A space $X$ can be filtered in several different ways. A multifiltration $X_\alpha$, for $\alpha$ a multi-index, is a collection of filtrations such that fixing all but one of the indices in $\alpha$ gives a (one-dimensional) filtration of $X$. The multidimensional persistence of $X_\alpha$ is a $|\alpha|$-dimensional grid of homology groups, with the barcode generalizing to the rank invariant, a map on the grid.
Another generalization, viewing filtrations as quivers, will not be discussed here, but rather presented as a separate post later.
References: Edelsbrunner and Morozov (Persistent homology: theory and practice), Carlsson, de Silva, and Morozov (Zigzag persistent homology and real-valued functions), Bubenik and Scott (Categorification of persistent homology), Carlsson and Zomorodian (The theory of multidimensional persistence)