A gentle introduction to persistent homology

Figure 2 illustrates this on a graph $G=(V,E)$ with a slightly adjusted filtration setup: Earlier, all points in our space represented their own connected component at step $m=0$. Let’s take a closer look at figure 2 to understand how a persistence diagram comes into existence:

Given a point $(x,y)$ from the persistence diagram $\mathcal{D}$, the persistence of the connected component which is represented by this tuple is defined as $$\text{pers}(x,y) := |y-x|. Furthermore, we can summarize these topological features of a persistence diagram by its $p$-norm: $$ ||\mathcal{D}||_p:= \Big(\sum_{(c,d)\in $$

I hope this post helped you to get an idea of what persistent homology is and you might already ponder about how to apply this to deep learning.

Source: christian.bock.ml