Cup Product Persistence and Its Efficient Computation

It is well-known that cohomology has a richer structure than homology. However, so far, in practice, the use of cohomology in persistence setting has been limited to speeding up of barcode computations. Two recently introduced invariants, namely, persistent cup-length and persistent Steenrod modules, to some extent, fill this gap. When added to the standard persistence barcode, they lead to invariants that are more discriminative than the standard persistence barcode. In this work, we introduce (the persistent variants of) the order-$k$ cup product modules, which are images of maps from the $k$-fold tensor products of the cohomology vector space of a complex to the cohomology vector space of the complex itself. We devise an $O(d n^4)$ algorithm for computing the order-$k$ cup product persistent modules for all $k \in \{2, \dots, d\}$, where $d$ denotes the dimension of the filtered complex, and $n$ denotes its size. Furthermore, we show that these modules are stable for Cech and Rips filtrations. Finally, we note that the persistent cup length can be obtained as a byproduct of our computations leading to a significantly faster algorithm for computing it.