Mapper on Manifolds: Adaptive Mapper Algorithms with Differential Geometry Priors

The review by Madukpe et al. (2025) highlights that Mapper’s power is often limited by the choice of cover and filter function—a largely heuristic process. This idea proposes a systematic integration of geometric priors (e.g., local curvature estimates, tangent space approximations) into Mapper’s construction. Instead of using standard covers (like intervals or balls), the algorithm would adaptively select regions that align with the manifold’s intrinsic geometry, potentially using diffusion maps or Laplacian eigenmaps as informed filters. This approach synthesizes TDA with advances from geometry and manifold learning, as hinted at in works like Moghadam & Pedram (2020), but pushes much further by making Mapper itself geometry-aware. The result would be Mapper graphs that better reflect true data topology, especially in high-noise or high-curvature regimes.

References:

1. Topological Data Analysis for Classification of DeepSat-4 Dataset. M. H. Moghadam, M. Pedram (2020). International Symposium on Telecommunications.
2. A Comprehensive Review of the Mapper Algorithm, a Topological Data Analysis Technique, and Its Applications Across Various Fields (2007-2025). Vine Nwabuisi Madukpe, B.C Ugoala, N. F. S. Zulkepli (2025).