Mapper

The idea of Mapper is to create a “compact summary” of point cloud data. We do this by covering the range of range of data with open sets, and then represent the points within each set with a small number of nodes representing clusters. We can then capture the intersections of each sets by taking the nerve.

Let $f:X\to {\mathbb{R}}^d$ be a continuous real-valued function, called a filter, and $\mathcal{U}=\{U_i{\}}_{i\in I}$ be an open cover of $f(X)$. The pullback cover of $X$ induced by $(f,\mathcal{U})$ is the collection of open sets $\{f^{-1}(U_i){\}}_{i\in I}$. The refined pullback is the collection of connected components of the open sets $f^{-1}(U_i)$, $i\in I$. The mapper complex $M(f,\mathcal{U})$ is defined as the nerve of the refined pullback.

Mapper Algorithm

Input: A set of points $X$ with a metric. Function $f:X\to {\mathbb{R}}^d$. A cover $\mathcal{U}$ of $f(X)$.

Steps:

1. For each $U_i\in \mathcal{U}$, decompose $f^{-1}(U_i)$ into clusters $C_{i,1},\dots ,C_{i,k_i}$
2. Compute the nerve of cover of $X$ defined by $\{C_{i,j}\}$.

Output: The simplicial complex.

The typical choice /is to use a filter $f:X\to \mathbb{R}$ and to cover $f(X)$ with uniform intervals. We call the length of the intervals the resolution and the percent overlap of intervals the gain.