Generalization

What is a mapper algorithm?

We can boil down the two mapper algorithms we saw earlier as follows:

1. (covering step) Given a metric space \((X, d)\), create a covering \(C\) of \(X\);
2. (nerve step) Using \(C\) as vertex set, create a graph.

In the classical mapper context, \(C\) is generated using the clustering of pre-images of a function \(f: X \to \mathbb{R}\). In the ball mapper scenario, we cover \(X\) using \(\epsilon\)-balls with centers as a subset of \(X\).

Covering step

Let \(X\), \(L\) and \(\epsilon\) be given as in the ball mapper case. For any \(l \in L\), define \(x_l\) = X[:, l]. Examples of how to generalize the covering step:

• Fix \(n > 0\) integer. Create a ball \(B\) of radius \(\epsilon\) around \(x_l\). If \(B\) contains less than \(n\) elements, then we redefine \(B\) as the set of \(n\) nearest neighbors of \(x_l\).

• Fix \(\lambda > 0\). Let \(d_l\) be the distance between \(x_l\) and its closest point. Create a ball \(B\) of radius \(\lambda * d_1\). Proceed like this to create a covering of \(X\).

Nerve step

There are many alternatives to the nerve construction. Let \(a\) and \(b\) be two elements of a covering \(C\). Let \(G = (V, E)\) be a graph with vertex-set \(V = C\). Examples of how to generalize the nerve step:

• Fix \(k > 0\). Define \((a, b) \in E\) iff \(|a \cap b| \geq k\), that is: we will only allow intersections with at least \(k\) elements. Setting \(k = 1\) will give us the usual nerve graph.

Creating my own mapper

You can change the 2 steps above within the context of the ball mapper, as can be seen in the docs.

doc

ball_mapper_generic(
    X::PointCloud, L::Vector{<:Integer}, 
    covering_function::Function,
    graph_function::Function
    )

Creates the ball mapper of a metric space X subsampled by L.

Arguments

• X::PointCloud: a point cloud.
• L::Vector{<:Integer}: a subset of index of X, that is: L is a subset of [1:size(X)[2]].
• covering_function::Function: a function that creates a cover for X. Its arguments are X and L.
• graph_function::Function: a function that creates a graph. It accepts a CoveredPointCloud object.

Details

See the “Generalization” page of the online documentation.