TDA Mapper Algorithm

The Mapper algorithm produces a topological summary of high-dimensional data. A filter function projects data onto R, overlapping intervals form a cover, clustering within each preimage creates nodes, and shared points create edges. The resulting graph captures the shape of the data.

Data Shape

Mapper Parameters

Mapper steps:
1. Apply filter f: X → R
2. Cover R with overlapping intervals {U_i}
3. Cluster each preimage f⁻¹(U_i)
4. Nodes = clusters; edges = shared points

Key property: Mapper is coordinate-free — only the topology matters. A circle gives a loop; a figure-8 gives two loops. The graph's Betti numbers (β₀ = components, β₁ = loops) encode the data's shape.