Graphical description of higher-dimensional Grassmannian persistence diagrams

Determine whether the degree-$\dgr$ Grassmannian persistence diagram obtained from the $\times$-Linear Orthogonal Inverse of birth-death spaces admits a useful graphical description for dimensions $\dgr\geq 0$, thereby extending the treegram interpretation beyond degree zero.

Background

The paper proves that degree-zero Grassmannian persistence diagrams are equivalent to treegrams, which provide a graphical representation of hierarchical clustering. This equivalence suggests that the higher-degree Grassmannian persistence diagrams may encode analogous structures, but the paper leaves unresolved whether those structures can be represented graphically in a useful way.

References

The equivalence of treegrams and degree-$0$ $\times$-Linear Orthogonal Inverses of birth-death spaces suggests that for dimensions $\dgr \geq 0$, $\prhi \left( \ZB_\dgr\Ffunc \right)$ can be thought of as a higher dimensional generalization of treegrams. This raises the question of whether there is a useful graphical description of $\prhi \left( \ZB_\dgr\Ffunc \right)$ in that case.

— Grassmannian Persistence Diagrams: Special Properties in the 1-Parameter Setting  (2504.06077 - Gülen et al., 4 Apr 2025) in Section Discussion