Coface Lifting (Simplicial to Combinatorial)

This technique lifts a Simplicial Complex to a Combinatorial Complex by taking the co-adjacencies of a simplex to be the components of the higher order ($3$-cell) cell. Given a simplex $\sigma$ and its co-faces $\tau_1, \tau_2, \cdot \cdot \cdot , \tau_i$. Then, using a purely combinatorial definition, $\delta = \sigma \cup \tau_1 \cup \cdot \cdot \cdot \cup \tau_i$. We can see that it holds that this is a combinatorial complex $\mathcal{X}$ because of the two conditions that need to be fulfilled.

1. All nodes $v$ are preserved. So given that the initial point cloud is denoted $S$ then it still holds that $\forall s \in S$ then {s} $\in \mathcal{X}$
2. If $\sigma,\delta \in \mathcal{X}$ and $\sigma \subseteq \delta$ then $rank(\sigma) \leq rank(\delta)$: This hold since we set $rank(\delta)$ to $3$ and we are operating on the a subset of the simplifies of a simplicial complex up to dimension $2$.

This technique is proposed in [1].

---

[1] Hajij, M., Zamzmi, G., Papamarkou, T., Miolane, N., Guzmán-Sáenz, A., Ramamurthy, K. N., ... & Schaub, M. T. (2022). Topological deep learning: Going beyond graph data. arXiv preprint arXiv:2206.00606.

---

Tags: Existing lift from literature | connectivity-based | deterministic

From https://github.com/pyt-team/challenge-icml-2024/pull/29