The Shift-Dimension of Multipersistence Modules

TL;DR

The paper defines the shift-dimension as a stable, algebraic invariant for multipersistence modules by applying hierarchical stabilization to the zeroth multigraded Betti number $\beta_0$. It develops a comprehensive framework using persistence contours to measure distances between multipersistence modules and proves stability properties, including a linear-time algorithm for computing $\dim_v$ in the bivariate interval-module case. It also extends the concept to multigraded $\mathbb{K}[x_1,\ldots,x_r]$-modules and introduces a versatile class of multivariate contours, linking topological data analysis to combinatorial commutative algebra. The results offer new tools for feature extraction in machine-learning contexts and pave the way for stabilizing higher Betti-number invariants and broader algebraic settings in multiparameter persistence.

Abstract

We present the shift-dimension of multipersistence modules and investigate its algebraic properties. This gives rise to a new invariant of multigraded modules over the multivariate polynomial ring arising from the hierarchical stabilization of the zeroth total multigraded Betti number. We give a fast algorithm for the computation of the shift-dimension of interval modules in the bivariate case. We construct multipersistence contours that are parameterized by multivariate functions and hence provide a large class of feature maps for machine learning tasks.

Figures (4)

• Figure 1: Illustration of a persistence module as in \ref{['ex:rectanglenonadditivity']} for $|I|=2$. Left: Let $v=(4,4).$ All shifted minimal generators have pairwise incomparable degrees but can be generated by the all-one vector in the degree of the greatest common divisor of their degrees. Hence, $\dim_v(\oplus_i(M_i))=1$. Right: The functions $\sum_i\dim_{\tau v}(M_i)$ (in lavender) and $\dim_{\tau v}(\oplus_i(M_i))$ (in blue) for $v=(1,1).$ We have $\mathop{\mathrm{Loc}}\nolimits_v ( \{ M_i\}_{i \in I}=[3,4.3)$ and $\mathop{\mathrm{err}}\nolimits_{v,p}=(0.6\cdot 2^p+0.7\cdot 3^p)^{1/p}.$
• Figure 2: In gray: the interval module $M$ of \ref{['example intervalmod']}. In lighter gray: $(4,4)\ast M.$Left: In blue and lavender: the submodules generated by the bases $\mathcal{B}_1$ and $\mathcal{B}_2,$ resp. Right: Regions in which the basis elements can be exchanged, fixing $(8,4)$ and $(4,6),$ resp., following \ref{['lemma:basis_exchange']}. Using this lemma again, both bases can be exchanged with $\mathcal{B}_3\coloneqq \{(4,6),(6,4)\}.$ Those replacements do not preserve the indispensability relations, since $(4,6)$ is indispensable only for ${(0,8)}$ and $(6,4)$ is indispensable only for $(11,0).$
• Figure 3: Left: Constructing a $(1.5,1.5)$-basis, using \ref{['algo']}, clustering from above. Right: Using the opposite version from below.
• Figure 4: Visualization of $M,x_1x_2M,$ and $x_1^2x_2^2M$ for $M$ from \ref{['example monomials']}

Theorems & Definitions (69)

• Definition 2.1
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7: GC
• Proposition 2.8: GC
• Example 2.9: Curve contour
• Example 2.10: Distance type in a fixed direction
• Remark 2.11