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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.
This technique is proposed in [1].
As additional contributions:
Dataset
and not aData
object.get_combinatorial_complex_connectivity
to generate connectivity matricesHMCModel
class that operates over combinatorial complexes.[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