Persistent homology

Remarks

A class is born at $i$ if it lies in $H_p(K_i; \F)$ but not in the image of $f_p^{i-1,i}$. It dies at $j$ if it merges with an older class or becomes $0$ when passing from $K_{j-1}$ to $K_j$. The persistence module splits, uniquely up to order, as a direct sum of interval modules. The list of their intervals $[b, d)$ is the barcode. The same data, as points $(b, d)$ in the plane, form the persistence diagram.

Example: let $K_0$ consist of two vertices $a, b$ and let $K_1$ add the edge $ab$. Then $H_0(K_0; \F) \cong \F^2$, $H_0(K_1; \F) \cong \F$, and $f_0^{0,1}$ is onto, so $\beta_0^{0,1} = 1$. The barcode in degree $0$ is $\set{[0, \infty), [0, 1)}$: one component lives forever and one dies when the edge joins them.

Non-example: a family of complexes $K_0, K_1$ where $K_1$ does not contain $K_0$ is not a filtration, and no maps $f_p^{0,1}$ are induced. Zigzag persistence treats such cases with maps in both directions.

For a finite set of points in a metric space, the standard filtrations are the Vietoris–Rips and Čech complexes at increasing scale.