Stabilizing decomposition of multiparameter persistence modules

TL;DR

This paper tackles the fundamental instability of decompositions in multiparameter persistence modules by introducing ε-refinements, ε-erosion neighborhoods, and ε-pruning to build a stability theory for approximate decompositions. The central result shows that when two modules are ε-interleaved, their pruning yields a common 2$r$ε-refinement, providing a stable, decompositional viewpoint across nearby modules, with $r$ the maximal pointwise dimension; this advances beyond naive interleaving-based notions by stabilizing decompositions rather than the modules themselves. It also introduces the erosion- and pruning-based distances $d_{EN}$ and $d_P$, analyzes their properties and limitations, and relates them to decompose-based conjectures, including a key conjecture connecting decompositions to graph-theoretic constraints (CI problems) in the upset/decomposition setting. The work clarifies when stable, decomposition-based metrics can exist (e.g., for interval-decomposable or upset-decomposable modules) and outlines a broader program linking stability, computation, and relative homological algebra in multipersistence. Collectively, the framework paves a practical path toward robust, informative summaries of multiparameter modules via stabilized prunings while highlighting deep, combinatorial barriers to universal stability.

Abstract

While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not been clear that there is any way to get around this and build a meaningful stability theory for multiparameter module decomposition. We introduce new tools, in particular $ε$-refinements and $ε$-erosion neighborhoods, to start building such a theory. We then define the $ε$-pruning of a module, which is a new invariant acting like a ``refined barcode'' that shows great promise to extract features from a module by approximately decomposing it. Our main theorem can be interpreted as a generalization of the algebraic stability theorem to multiparameter modules up to a factor of $2r$, where $r$ is the maximal pointwise dimension of one of the modules. Furthermore, we show that the factor $2r$ is close to optimal. Finally, we discuss the possibility of strengthening the stability theorem for modules that decompose into pointwise low-dimensional summands, and pose a conjecture phrased purely in terms of basic linear algebra and graph theory that seems to capture the difficulty of doing this. We also show that this conjecture is relevant for other areas of multipersistence, like the computational complexity of approximating the interleaving distance, and recent applications of relative homological algebra to multipersistence.

Figures (8)

• Figure 1: Illustration of \ref{['thm_main']} (ii), which says that if $M$ and $N$ are $\epsilon$-interleaved, then the $\epsilon$-pruning of $M$ is a $2r\epsilon$-refinement of $N$, where $r$ is the maximum pointwise dimension of $M$. If \ref{['conj_main']} holds, then the same statement holds if $r$ is the maximum pointwise dimension of the modules in $B(M)$. (That is, the $M_i$ in the figure.)
• Figure 2: The module $M= I\oplus I'$ on the left, $N= J\oplus J'$ in the middle, and $Q = J\oplus K\oplus I'$ on the right. $M$ and $N$ are $0.03$-interleaved and have a common $0.03$-refinement $Q$, but any bijection between $\{I, I'\}$ and $\{J, J'\}$ would match two components that are not $\delta$-interleaved for any $\delta<0.4$.
• Figure 3: Left: The direct sum $M$ of two interval modules supported on $[0.8,3]^2$. Middle left: an indecomposable module $N$. Middle right: The $0.5$-erosion of $N$, which is also indecomposable. Right: The $0.1$-pruning $\mathop{\mathrm{Pru}}\nolimits_{0.1}(M)$ of $M$ decomposes into two summands, allowing a good matching with the components of $M$. It turns out that $\mathop{\mathrm{Pru}}\nolimits_{0.1}(M)$ is a $0.3$-refinement of both $M$ and $N$.
• Figure 4: An illustration of the relationships between the subquotients of $M$, the interleaving and erosion neighborhoods of $M$, and the $\epsilon$-refinements and $\epsilon$-erosion of $M$. The figure is somewhat inconsistent for easier visualization: $\mathop{\mathrm{EN}}\nolimits_\epsilon(M)$ is a set of equivalence classes of modules up to isomorphism. Since $\mathbf{Er}_\epsilon(M)$ is a set of isomorphic modules, it only represents one element of $\mathop{\mathrm{EN}}\nolimits_\epsilon(M)$, contrary to how it is drawn.
• Figure 5: The black curve is the boundary of the support of $M$. A module isomorphic to $N$ is drawn in red, with the shade illustrating the dimension in the four constant regions of the support.

Theorems & Definitions (115)

• Corollary
• Definition 2.1
• Theorem 2.2: botnan2020decomposition, azumaya1950corrections
• Definition 2.3
• Definition 2.4
• Theorem 2.5: The algebraic stability theorem, chazal2009proximity
• Definition 2.6
• Definition 3.1
• Definition 3.2
• Remark 3.3