The discriminating power of the generalized rank invariant

The rank invariant (RI), one of the best known invariants of persistence modules $M$ over a given poset $P$, is defined as the map sending each comparable pair $p\leq q$ in $P$ to the rank of the linear map $M(p\leq q)$.The recently introduced notion of generalized rank invariant (GRI) acquires more discriminating power than the RI at the expense of enlarging the domain of RI to the set Int$(P)$ of intervals of $P$ or to the even larger set Con$(P)$ of connected subposets of $P$. Given that the size of Int$(P)$ and Con$(P)$ can be much larger than the domain of the RI, restricting the domain of the GRI to smaller, more manageable subcollections $\mathcal{I}$ would be desirable to reduce the total cost of computing the GRI. This work studies the tension between computational efficiency and strength when restricting the domain of the GRI to different choices of $\mathcal{I}$. In particular, we prove that in terms of discriminating power, the GRI over restricted collections $\mathcal{I}$ faithfully interpolates between the RI and the GRI over Int$(P)$. We also establish that for suitable collections $\mathcal{I}$, the GRI over $\mathcal{I}$ is stable. Finally, we introduce the notion of Zigzag-path-Indexed Barcode (ZIB) for persistence modules $M$ over a finite 2d-grid, which is a function that sends each zigzag path $\Gamma$ in the 2d-grid to the barcode of the restriction of $M$ to $\Gamma$. Since the RI is equivalent to the fibered barcode (i.e. the ZIB induced by monotone paths), the ZIB is a natural refinement of the RI. Motivated by a recent finding that zigzag persistence can be used to compute the GRI of $M$, we compare the discriminating power of the ZIB with that of the GRI. Clarifying the connection between the GRI and the ZIB is necessary to understand to what extent zigzag persistence algorithms can be exploited for computing the GRI.