Added subsection 8.1.2

rmd/08-hasse-graph-interpretation.rmd

@@ -37,3 +37,13 @@ The proof of the three parts of the proposition follows by noting that the struc
 ```{r vis-structure, echo=FALSE, fig.align="center", fig.cap="Relation between immediate incidence and (co)adjacency on the Hasse graph of a CC. (a): Two $(k-1)$ cells $x^{k-1}$ and $y^{k-1}$ (orange vertices) are 1-adjacent if and only if there exists a $k$-cell $z^k$ (pink vertex) that is incident to $x^{k-1}$ and $y^{k-1}$. (b): Two $(k+1)$ cells $x^{k+1}$ and $y^{k+1}$ (blue vertices) are 1-coadjacent if and only if there exists a $k$-cell $z^k$ (pink vertex) that is incident to $x^{k+1}$ and $y^{k+1}$."}
 knitr::include_graphics('figures/prop_structure.png', dpi=NA)
 ```
+
+### Augmented Hasse graphs
+
+The Hasse graph of a CC is useful because it shows that computations for a higher-order deep learning model can be reduced to computations for a graph-based model. Particularly, a $k$-cochain (signal) being processed on a CC $\mathcal{X}$ can be thought as a signal on the corresponding vertices of the associated Hasse graph $\mathcal{H}_{\mathcal{X}}$. The edges specified by the matrices $B_{k,k+1}$ determine the message-passing structure of a given higher-order model defined on $\mathcal{X}$. However, the message-passing structure determined via the matrices $A_{r,k}$ is not directly supported on the corresponding edges of $\mathcal{H}_{\mathcal{X}}$. Thus, it is sometimes desirable to *augment the Hasse graph* with additional edges other than the ones specified by the poset partial order relation of the CC. Along these lines, Definition \@ref(def:ahg) introduces the notion of augmented Hasse graph.
+
+```{definition, ahg, name="Augmented Hasse graph"}
+Let $\mathcal{X}$ be a CC, and let $\mathcal{H}_{\mathcal{X}}$ be its Hasse graph with vertex set $V(\mathcal{H}_{\mathcal{X}})$ and edge set $E(\mathcal{H}_{\mathcal{X}})$. Let $\mathcal{N}=\{\mathcal{N}_1,\ldots,\mathcal{N}_n\}$ be a set of neighborhood functions defined on $\mathcal{X}$. We say that $\mathcal{H}_{\mathcal{X}}$ has an augmented edge $e_{x,y}$ induced by $\mathcal{N}$ if there exist $\mathcal{N}_i \in \mathcal{N}$ such that $x \in \mathcal{N}_i(y)$ or $y \in \mathcal{N}_i(x)$. Denote by $E_{\mathcal{N}}$ the set of all augmented edges induced by  $\mathcal{N}$. The *augmented Hasse graph* of $\mathcal{X}$ induced by $\mathcal{N}$ is defined to be the graph $\mathcal{H}_{\mathcal{X}}(\mathcal{N})= (V(\mathcal{H}_{\mathcal{X}}), E(\mathcal{H}_{\mathcal{X}}) \cup E_{\mathcal{N}})$.
+```
+
+It is easier to think of the augmented Hasse graph in Definition \@ref(def:ahg) in terms of the matrices $\mathbf{G}=\{G_1,\ldots,G_n\}$ associated with the neighborhood functions $\mathcal{N}=\{\mathcal{N}_1,\ldots,\mathcal{N}_n\}$. Each augmented edge in $\mathcal{H}_{\mathcal{X}}(\mathcal{N})$ corresponds to a non-zero entry in some $G_i\in \mathbf{G}$. Since $\mathcal{N}$ and $\mathbf{G}$ store equivalent information, we use $\mathcal{H}_{\mathcal{X}}(\mathbf{G})$ to denote the augmented Hasse graph induced by the edges determined by $\mathbf{G}$. For instance, the graph given in Figure \@ref(fig:hasse-diagram)(c) is denoted by $\mathcal{H}_{\mathcal{X}}( A_{0,1},coA_{2,1})$.