A Glossary
The table below summarizes the paper’s notation.
| Notation | Description |
|---|---|
| \(S\) | Non-empty finite set of abstract entities |
| \(P_{S}\) | Index set |
| \(\mathcal{P}(S)\) | Power set of a set \(S\) |
| \((S,\mathcal{N})\) | Topological space of a nonempty set \(S\) and a neighborhood topology \(\mathcal{N}\) |
| \(\mathcal{N}_{a}(x)\) | Adjacency set of a cell \(x\) |
| \(\mathcal{N}_{co}(x)\) | Coadjacency set of a cell \(x\) |
| \(\mathcal{N}_{\{ G_1,\ldots,G_n\}}(x)\) | Neighbors of \(x\) specified by the neighborhood matrices \(\{G_1,\dots,G_n\}\) |
| \(\mathcal{N}_{\searrow}(x)\) | Set of down-incidence of a cell \(x\) |
| \(\mathcal{N}_{\nearrow}(x)\) | Set of up-incidence of a cell \(x\) |
| \(\mathcal{N}_{\searrow,k}(x)\) | Set of \(k\)-down incidence of a cell \(x\) |
| \(\mathcal{N}_{\nearrow,k}(x)\) | Set of \(k\)-up incidence of a cell \(x\) |
| \(\mathbb{N}\) and \(\mathbb{Z}_{\ge 0}\) | Set of positive integers and non-negative integers, respectively |
| \(\mathcal{G}\) | Graph |
| \(x^k\) | Cell \(x\) of rank \(k\) |
| \(\mbox{rk}\) | Rank function |
| \((S, \mathcal{X}, \mbox{rk})\) | CC, consisting of a set \(S\), a subset \(\mathcal{X}\) of \(\mathcal{P}(S)\setminus\{\emptyset\}\), and a rank function \(\mbox{rk}\) |
| \(\dim (\mathcal{X})\) | Dimension of a CC \(\mathcal{X}\) |
| \(\{c_\alpha\}_{\alpha \in I}\) | Partition into subspaces (cells) indexed by an index set \(I\) |
| \(\mbox{int}(x)\) | Interior of a cell \(x\) in a regular cell complex |
| \(n_\alpha\in \mathbb{N}\) | Dimension of a cell in a regular cell complex |
| \(0\)-cells | Vertices of a CC |
| \(1\)-cells | Edges of a CC |
| \(k\)-cells | Cells with rank \(k\) |
| \(\mathcal{X}^{(k)}\) | \(k\)-skeleton of \(\mathcal{X}\), formed by \(i\)-cells in \(\mathcal{X}\) with \(i\leq k\) |
| \(\mathcal{X}^k\) | Set of k-cells of \(\mathcal{X}\) |
| \(|\mathcal{X}^k|\) | Cardinality of \(\mathcal{X}^k\), that is number of \(k\)-cells of \(\mathcal{X}\) |
| \(\mathcal{X}_{n-hop}(G)\) | \(n\)-hop CC of a graph \(G\) |
| \(\mathcal{X}_p(G)\) | Path-based CC of a graph \(G\) |
| \(\mathcal{X}_{loop}(G)\) | Loop-based CC of a graph \(G\) |
| \(\mathcal{X}_{SC}(\mathcal{Y})\) | Coface CC of a simplicial complex/CC \(\mathcal{Y}\) |
| \(B_{r,k}\) | Incidence matrices between \(r\)-cells and \(k\)-cells |
| \(A_{r,k}\) | Adjacency matrices among the cells of \(\mbox{X}^{r}\) with respect to the cells of \(\mbox{X}^{k}\) |
| \(coA_{r,k}\) | Coadjacency matrices among the cells of \(\mbox{X}^{r}\) with respect to the cells of \(\mbox{X}^{k}\) |
| \(\mathbf{W}\) | Trainable parameter |
| \(\mathcal{C}^k(\mathcal{X},\mathbb{R}^d)\) | \(k\)-cochain space with features in \(\mathbb{R}^d\) |
| \(\mathcal{C}^k\) | \(k\)-cochain space with features in some Euclidean space |
| \(\mathbf{G}= \{G_1,\ldots,G_m\}\) | Set of cochain maps \(G_i\) defined on a complex |
| \(\mathcal{M}_{ \mathbf{G};\mathbf{W}}\) | Merge node |
| \(G:C^{s}(\mathcal{X})\to C^{t}(\mathcal{X})\) | Cochain map |
| \((\mathbf{x}_{i_1},\ldots, \mathbf{x}_{i_m})\) | Vector of cochains |
| \(att^{l}: C^{s}(\mathcal{X})\to C^{s}(\mathcal{X})\) | Higher-order attention matrix |
| \(\mathcal{N}_{\mathcal{Y}_0}=\{\mathcal{Y}_1,\ldots,\sigma_{|\mathcal{N}_{\mathcal{Y}_0}|}\}\) | Set of a complex object in the vicinity of \(\mathcal{Y}_0\) |
| \(a: {\mathcal{Y}_0}\times \mathcal{N}_{\mathcal{Y}_0}\to [0,1]\) | Higher-order attention function |
| \(\mbox{CCNN}_{\mathbf{G};\mathbf{W}}\) | CCNN or its tensor diagram representation |
| \(\mathcal{H}_{\mathcal{X}}= (V (\mathcal{H}_{\mathcal{X}}), E(\mathcal{H}_{\mathcal{X}}) )\) | Hasse graph with vertices \(V (\mathcal{H}_{\mathcal{X}})\) and edges \(E(\mathcal{H}_{\mathcal{X}})\); see Definition 8.1 |
The table below summarizes the paper’s acronyms.
| Acronym | Description |
|---|---|
| AGD | Average geodesic distance |
| CC | Combinatorial complex |
| CCANN | Combinatorial complex attention neural network |
| CCCNN | Combinatorial complex convolutional neural network |
| CCNN | Combinatorial complex neural network |
| CNN | Convolutional neural network |
| DEC | Discrete exterior calculus |
| GDL | Geometric deep learning |
| GNN | Graph neural network |
| MOG | Mapper on graphs |
| RNN | Recurrent neural network |
| SCoNe | Simplicial complex network |
| Sub-CC | sub-combinatorial complex |
| TDA | Topological data analysis |
| TDL | Topological deep learning |
| TQFT | Topological quantum field theory |