Structural Liftings

In the context of TDL, liftings facilitate the mapping of data from one topological representation to another. This mapping comprises two key aspects: structural lifting and feature lifting.

• The structural lifting is responsible for the transformation of the underlying relationships or elements of the data. For instance, it might determine how nodes and edges in a graph are mapped into triangles and tetrahedra in a simplicial complex. This structural transformation can be further categorized into connectivity-based, where the mapping relies solely on the existing connections within the data, and feature-based, where the data's inherent properties or features guide the new structure.
• The feature lifting, conversely, addresses the transfer of data attributes or features during mapping, ensuring that the properties associated with the data elements are consistently preserved in the new representation.

This wiki describes in detail the structural liftings implemented in TopoBench --taxonomized by source and destination topological domains. Most of them are submissions of the ICML TDL Challenge 2024 (paper | repo):

Graph to Simplicial Complex

Name	Type	Description
DnD Lifting		Wiki page
Random Latent Clique Lifting		Wiki page
Line Lifting		Wiki page
Neighbourhood Complex Lifting		Wiki page
Graph Induced Lifting		Wiki page
Clique Lifting	Connectivity-based	Wiki page
K-hop Lifting	Connectivity-based	Wiki page

Graph to Cell Complex

Name	Type	Description
Discrete Configuration Complex		Wiki page
Cycle Lifting	Connectivity-based	Wiki page

Graph to Hypergraph

Name	Type	Description
Expander Hypergraph Lifting		Wiki page
Kernel Lifting		Wiki page
Mapper Lifting		Wiki page
Forman‐Ricci Curvature Coarse Geometry Lifting		Wiki page
KNN Lifting		Wiki page
K-hop Lifting	Connectivity-based	Wiki page

Graph to Combinatorial Complex

Name	Type	Description
Matroid lifting		Wiki page
Molecule Ring & Close Atoms Lifting		Wiki page
Simplicial Paths Lifting		Wiki page

Pointcloud to Simplicial

Name	Type	Description
Delaunay Lifting		Wiki page
Random Flag Complex		Wiki page

Pointcloud to Hypergraph

Name	Type	Description
Mixture of Gaussians MST lifting		Wiki page
PointNet Lifting		Wiki page
Voronoi Lifting		Wiki page

Hypergraph to Simplicial

Name	Type	Description
Hypergraph Heat Kernel Lifting		Wiki page

Hypergraph to Combinatorial

Name	Type	Description
Universal Strict Lifting		Wiki page

• Graph to Simplicial Complex
  • DnD Lifting
  • Random Latent Clique Lifting
  • Line Lifting
  • Neighbourhood Complex Lifting
  • Graph Induced Lifting
• Graph to Cell Complex
  • Discrete Configuration Complex
• Graph to Hypergraph
  • Expander Hypergraph Lifting
  • Kernel Lifting
  • Mapper Lifting
  • Forman‐Ricci Curvature Coarse Geometry Lifting
  • KNN Lifting
• Graph to Combinatorial
  • Matroid lifting
  • Molecule Ring & Close Atoms Lifting
  • Simplicial Paths Lifting
• Pointcloud to Simplicial
  • Delaunay Lifting
  • Random Flag Complex
• Pointcloud to Hypergraph
  • Mixture of Gaussians MST lifting
  • PointNet Lifting
  • Voronoi Lifting
• Hypergraph to Simplicial
  • Hypergraph Heat Kernel Lifting
• Hypergraph to Combinatorial
  • Universal Strict Lifting