Computation of $γ$-linear projected barcodes for multiparameter persistence

Abstract

The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules along linear forms in the polar of some fixed cone $\gamma$. So far, the computation of the $\gamma$-linear projected barcode has only been studied in the functional setting, in which persistence modules come from the persistent cohomology of $\mathbb{R}^n$-valued functions. Here we develop a method that works in the algebraic setting directly, for any multiparameter persistence module over $\mathbb{R}^n$ that is given via a finite free resolution. Our approach is similar to that of RIVET: first, it pre-processes the resolution to build an arrangement in the dual of $\mathbb{R}^n$ and a barcode template in each face of the arrangement; second, given any query linear form $u$ in the polar of $\gamma$, it locates $u$ within the arrangement to produce the corresponding barcode efficiently. While our theoretical complexity bounds are similar to the ones of RIVET, our arrangement turns out to be simpler thanks to the linear structure of the space of linear forms. Our theoretical analysis combines sheaf-theoretic and module-theoretic techniques, showing that multiparameter persistence modules can be converted into a special type of complexes of sheaves on vector spaces called conic-complexes, whose derived pushforwards by linear forms have predictable barcodes.