Topologies Definition:

Simplical Complexes

Graphes as 1D complex

0-simplex: vertices

1-simplex: edges

→ pairwise relationships

Triangles: 2D complex

0-simplex: vertices

1-simplex: edges

2-simplex: filled triangles

→ 3 way relationships

Tetrahedron: 3D complex (4 points pyramids)

0-simplex: vertices

1-simplex: edges

2-simplex: faces / filled triangles

3-simplex: volume

→ 4 way relationships

Cell Complexes: discrete representations, type-set hierarcht

cell complexes are a generatlization of simplical complexes that follows an interior to boundary hierarchy, and a not limited to complexes structures i.e faces can involve more than three nodes,

0-rank cell: nodes

1-rank cell: edges / bonds

2-rank cell: rings

3-rank cell: cylinder

Combinatorial Complexes: continuous representations interior-boundary structure

(generalization final boss)

A Combinatorial Complex is a triplet $(\mathcal{V}, \mathcal{X}, \mathcal{rk})$ where

$\mathcal{V}$ is a set of vertics

$\mathcal{X} \subset \mathcal{P}(\mathcal{V})$ a subset of the powerset of $\mathcal{V}$

$\mathcal{rk}$ an order preserving rank function

Topological Lifting

let $\mathcal{T_1} = (\mathcal{F_1}, \mathcal{X_1})$ and $\mathcal{T_2} = (\mathcal{F_2}, \mathcal{X_2})$ two featured topological domains

a lifting from $\mathcal{T_1}$ to $\mathcal{T_2}$ is a pair $(\mathcal{\psi}\mathcal{X}, \mathrm{\psi}\mathrm{F})$ where:

$$ \text{Structural lifting:} \quad \quad \quad \quad \quad\mathcal{\psi}_\mathrm{X}: \mathcal{X}_1 \times \mathcal{F}_1 \rightarrow \mathcal{X}_2 \\

\text{feature lifting: \quad\quad\quad\quad\quad\quad }\mathcal{\psi}_\mathrm{F}: \mathcal{X}_1 \times \mathcal{F}_1 \rightarrow \mathcal{F}_2 \\

\text{preserving: } \quad\quad\quad\quad\quad \mathrm{F}2(\mathcal{\psi}\mathrm{X}(x)) = \mathcal{\psi}_\mathrm{F}(\mathrm{F}_1(x)) $$

Topological Neural Nets in k-cochain spaces

k-Cochain spaces:

let $\mathcal{C}^k (\mathcal{X}, \mathbb{R}^d)$ be a $\mathbb{R}$-vector space of functions $\mathrm{H}_k$ where $\mathrm{H}_k: \mathcal{X}^k \rightarrow \mathbb{R}^d$

elements $\mathrm{H}_k$ are the k-cochain (k-signals) of the k-cochain space $\mathcal{C}^k (\mathcal{X}, \mathbb{R}^d)$

TNN: topological neural nets

a topological neural network is function mapping a set of k-signals to another

$$ \mathrm{TNN}: \mathcal{C}^{i_1} \times \dots \times \mathcal{C}^{i_n} \rightarrow \mathcal{C}^{j_1} \times \dots \times \mathcal{C}^{j_m} $$

Mathematical Modeling of high-order Complexes

cochains are structured through inter-domain and intra-domain mappings, the adjacency and incidence matrices respectively

formally defined as

for r < k, incidence matrices

$$ \mathrm{B}{r, k}: \mathcal{C}^k(\mathcal{X}) \rightarrow \mathcal{C}^r(\mathcal{X})\\ \mathrm{A}{r, k}: \mathcal{C}^r(\mathcal{X}) \rightarrow \mathcal{C}^r(\mathcal{X}) $$

incidence and adjacency matrices collectively redistribute signals across different dimensional cells

Neighborhood function: $\mathcal{N}: \mathrm{S} \rightarrow \mathcal{P}(\mathcal{P}(\mathrm{S}))$

HOMP: Higher-Order Message Passing

let $\mathcal{X}$ be a topological domain, let $\mathcal{N}(x)$ denote multi-set $\{\!\{\mathcal{N_1(x)}, \dots, \mathcal{N}_n(x)\}\!\}$ be a set of neighborhood functions defined on $\mathcal{X}$

given $y \in \mathcal{N}(x)$ and $h^{(l)}_x$ the feature vector of the element $x$ HOMP is defined as:

$$ m_{x, y} = \alpha_{\mathcal{N}_x} (h^{(l)}x, h^{(l)}y) \\ m^k_x = \bigoplus{y \in \mathcal{N}k(x)} m{x, y}, 1 \le k \le n \\ m_x = \bigotimes{\mathcal{N}_k \in \mathcal{N}(x)} m^k_x\\ h^{(l+1)}_k = \beta(h^{(l)}_x, m_x) $$

Read out layers/ techniques

Definitions and acronyms

posets: partially ordered sets