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This question relates to the topic of Persistent Homology, a branch of (applied) algebraic topology.

In the paper (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.116.2471&rep=rep1&type=pdf, page 6), the "$p$-persistent $k$th homology group" is defined as: $$H_k^{i,p}=Z_k^i/(B_k^{i+p}\cap Z_k^i)$$

While I understand the notation, and also understand the alternative interpretation as the homomorphism $\eta_k^{i,p}: H_k^i\to H_k^{i+p}$ that maps a homology class into the one that contains it, I don't really grasp the intuitive meaning behind it.

For homology group, for example, intuitively we understand that the $n$th homology group counts the number of $n$-dimensional "holes". What then, does the persistent group actually mean? (After reading many papers, I roughly get the idea that it measures the holes that "persist" but my understanding remains vague)

Q2) For the next question, what does the persistent module (defined by Carlsson in page 6 of the same paper) actually mean and how is it related to the persistent group? Again, I vaguely get the idea that the "persistent homology of a filtered complex is standard homology of a graded module over a polynomial ring", but fail to grasp why is this important/useful.

Thanks for any explanation! (Any reference will be greatly appreciated too.)

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Regarding the homology $H^{i,p}_k$ for the filtration
$$0 = K^0 \subseteq K^1 \subseteq \cdots \subseteq K^m =K $$we look at the persistent complex around $K^i_k$: $$\begin{array}{cccccccccc} & & \Big\downarrow \small\partial^i_{k+2} & & \Big\downarrow \small\partial^{i}_{k+2} & & & & \Big\downarrow \small\partial^{i+p}_{k+2} & & &\\ \cdots & \xrightarrow{\eta^{i-1}} & K^{i}_{k+1} & \xrightarrow{\eta^i} & K^{i+1}_{k+1} & \xrightarrow{\eta^{i+1}} & \cdots & \xrightarrow{\eta^{i+p-1}}& K^{i+p}_{k+1} & \xrightarrow{\eta^{i+p}} & \cdots\\ & & \Big\downarrow \small\partial^i_{k+1} & & \Big\downarrow \small\partial^{i+1}_{k+1} & & & & \Big\downarrow \small\partial^{i+p}_{k+1} & & &\\ \cdots & \xrightarrow{\eta^{i-1}} & K^{i}_{k} & \xrightarrow{\eta^i} & K^{i+1}_{k} & \xrightarrow{\eta^{i+1}} & \cdots & \xrightarrow{\eta^{i+p-1}}& K^{i+p}_{k} & \xrightarrow{\eta^{i+p}} & \cdots\\ & & \Big\downarrow \small\partial^i_{k} & & \Big\downarrow \small\partial^{i+1}_{k} & & & & \Big\downarrow \small\partial^{i+p}_{k} & & &\\ \cdots & \xrightarrow{\eta^{i-1}} & K^{i}_{k-1} & \xrightarrow{\eta^i} & K^{i+1}_{k-1} & \xrightarrow{\eta^{i+1}} & \cdots & \xrightarrow{\eta^{i+p-1}}& K^{i+p}_{k-1} & \xrightarrow{\eta^{i+p}} & \cdots\\ & & \Big\downarrow \small\partial^i_{k-1} & & \Big\downarrow \small\partial^{i+1}_{k-1} & & & & \Big\downarrow \small\partial^{i+p}_{k-1} & & &\\ \end{array} $$ If we fix $i$ we can take the usual homology on the complex $K^i_*$ with $H^i_k=Z^i_k/B^i_k$ where we have $$Z^i_k \approx ker \, \partial^i_k \,\, \text{and} \,\, B^i_k \approx Im \, \partial^i_{k+1}$$ Intuitively this is, $Z^i_k \subset K^i_k$ are the cycles in $K^i_k$ and $B^i_k \subset K^i_k$ are the elements of $K^i_k$ which are the boundaries of some $\gamma \in K^i_{k+1}$. Thus $H^i_k$ are the elements of $K^i_k$ which are cycles but not boundaries of elements in $K^i_{k+1}$.

Similarly for $H^{i,p}_k \approx Z^i_k / (B^{i+p}_k \cap Z^i_k)$ $Z^i_k$ are cycles in $K^i_k$ and $B^{i+p}_k$ are the boundaries of elements $\gamma \in K^{i+p}_{k+1}$ and consequently boundaries of $\sigma \in K^i_{k+1}$. So elements of $H^{i+p}_k$ are cycles i $K^i_k$ which are not the boundaries for any element $\gamma \in K^j_{k+1}; \forall j \leq i+p$. In this way $H^{i,p}_k$ is a further restriction on $H^i_k$.

Regarding your second question. As stated in the paper the above filtered complex is a persistent complex. To obtain the desired persistent module for fixed $i$ we compute $H^i_k$ (not the persistent homology) for all $k$ then take the sum $\bigoplus_{k=0}^\infty H^i_k$ and with the inclusion maps $\eta^{i}: \bigoplus_{k=0}^\infty H^i_k \rightarrow \bigoplus_{k=0}^\infty H^{i+1}_k$ we have the desired persistence module. As to how this module is connected to the persistent homology group, see the section of the source 3.3 Interpretation in which the authors ask "when $e+B^l_k$ is a basis element for the persistent groups $H^{l,p}_k$."

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