Topological Data Analysis — Mapper Algorithm Graph

Singh-Mémoli-Carlsson Mapper (2007): compress high-dimensional data into a topological skeleton graph revealing loops, flares, clusters.

Data cloud (2D projection)
Mapper graph (topological summary)

Mapper Parameters

Nodes: — | Edges: —
Mapper (2007): 1) choose filter f: X→ℝ. 2) Cover f(X) with overlapping intervals. 3) For each interval, cluster preimage f⁻¹(Uᵢ). 4) Connect clusters sharing points. The result is a graph that captures topology: loops → circles, branches → flares.