Asymptotic Dimension of Persistence Modules of Bounded Presentation
Sushovan Majhi, Atish Mitra, Žiga Virk
原文摘要
We compute the asymptotic dimension of spaces of multiparameter persistence modules of bounded presentation under the interleaving distance. The $d$-parameter modules with at most $n$ generators and $m$ relations have asymptotic dimension, and Assouad-Nagata dimension, exactly $d(n+m)$, the number of coordinates of their generator and relation grades, over every field; for two parameters this is $2(n+m)$. The space has continuous moduli, already for two parameters and $n,m\ge2$: over an infinite field it contains, at every scale, infinite uniformly separated families of pairwise non-isomorphic modules with one fibered barcode, so it is not doubling even at a fixed fibered barcode; already for $m\ge1$ it is not coarsely equivalent to a proper metric space. The proof is an induction on the number of distinct grade coordinates: an interleaving small compared with the gaps between these coordinates moves each of them by at most twice its size and preserves the induced grid representation, so the continuous moduli are invisible at large scale. Each class of modules at finite distance therefore embeds coarsely into Hilbert space; for $n,m\ge2$ over an uncountable field, the classes that contain these families admit no such embedding into a separable Hilbert space. Under the two-parameter matching distance $2(n+m)$ is a lower bound, attained for one generator and for two generators with at most one relation; on a finite net of $S$ slices the asymptotic dimension is at most $2nS$.
math.MG · cs.CG · math.AT