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 currently compatible with TopoBench --taxonomized by source and destination topological domains. Most of them are winner submissions of the ICML TDL Challenge 2024 (paper | repo), and a few of them are still in progress of being implemented in TB:

Graph to Simplicial Complex

Name	Type	Description
DnD Lifting	Feature-based	Wiki page
Random Latent Clique Lifting	Connectivity-based	Wiki page
Line Lifting	Connectivity-based	Wiki page
Neighbourhood Complex Lifting	Connectivity-based	Wiki page
Graph Induced Lifting	Connectivity-based	Wiki page
Eccentricity Lifting	Connectivity-based	Wiki page
Feature‐Based Rips Complex	Both connectivity and feature-based	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	Connectivity-based	Wiki page
Cycle Lifting	Connectivity-based	Wiki page

Graph to Hypergraph

Name	Type	Description
Expander Hypergraph Lifting	Connectivity-based	Wiki page
Kernel Lifting	Both connectivity and feature-based	Wiki page
Mapper Lifting	Connectivity-based	Wiki page
Forman‐Ricci Curvature Coarse Geometry Lifting	Connectivity-based	Wiki page
Protein Close Residues Lifting	Both connectivity and feature-based	Wiki page
Modularity Maximization Lifting	Both connectivity and feature-based	Wiki page
KNN Lifting	Feature-based	Wiki page
K-hop Lifting	Connectivity-based	Wiki page

Graph to Combinatorial Complex

Name	Type	Description
Matroid lifting	Connectivity-based	Wiki page
Molecule Ring & Close Atoms Lifting	Both connectivity and feature-based	Wiki page
Molecule Ring & Functional Lifting	Both connectivity and feature-based	Wiki page
Simplicial Paths Lifting	Connectivity-based	Wiki page

Pointcloud to Graph

Name	Type	Description
Protein Lifting	Feature-based	Wiki page

Pointcloud to Simplicial

Name	Type	Description
Delaunay Lifting	Feature-based	Wiki page
Random Flag Complex	Feature-based	Wiki page

Pointcloud to Hypergraph

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

Simplicial to Combinatorial

Name	Type	Description
Coface Lifting	Connectivity-based	Wiki page

Hypergraph to Simplicial

Name	Type	Description
Hypergraph Heat Kernel Lifting	Connectivity-based	Wiki page

Hypergraph to Combinatorial

Name	Type	Description
Universal Strict Lifting	Connectivity-based	Wiki page