On the stability of multigraded Betti numbers and Hilbert functions

TL;DR

This work addresses bottleneck stability for multigraded Betti numbers by introducing signed Betti barcodes and a balanced bottleneck dissimilarity. It proves a universal stability bound $\widehat{d_B}(\beta\mathcal{B}(M),\beta\mathcal{B}(N)) \le (n^2-1)\,d_I(M,N)$, derived from Hilbert’s syzygy theorem, free-module stability, and projective-resolution stability, and extends a $1$-Wasserstein stability result to the $1$- and $2$-parameter cases with $\widehat{d_{W^1}}(\mathcal{HB}(M),\mathcal{HB}(N)) \le n\,d_I^1(M,N)$. It also analyzes the existence/uniqueness and instability of minimal Hilbert decompositions, and provides algorithmic methods for computing Betti numbers, Hilbert decompositions, and dissimilarities. The paper further derives stability consequences for Hilbert functions, sublevel-set Betti numbers, and invariants from exact structures, and discusses universality, topologies, and practical implications for multiparameter persistence analysis. Overall, it offers a rigorous, computable stability framework that ties algebraic invariants to interleaving-distance geometry with potential applications in statistics and machine learning on persistence data.

Abstract

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free modules in its minimal free resolution -- as well as in practice -- it is easy to visualize and it is one of the main outputs of current multiparameter persistent homology software, such as RIVET. However, to the best of our knowledge, no bottleneck stability result with respect to the interleaving distance has been established for this invariant so far, and this potential lack of stability limits its practical applications. We prove a stability result for multigraded Betti numbers, using an efficiently computable bottleneck-type dissimilarity function we introduce. Our notion of matching is inspired by recent work on signed barcodes, and allows matching bars of the same module in homological degrees of different parity, in addition to matchings bars of different modules in homological degrees of the same parity. Our stability result is a combination of Hilbert's syzygy theorem, Bjerkevik's bottleneck stability for free modules, and a novel stability result for projective resolutions. We also prove, in the $2$-parameter case, a $1$-Wasserstein stability result for Hilbert functions with respect to the $1$-presentation distance of Bjerkevik and Lesnick.

Figures (4)

• Figure 1: The multigraded Betti numbers of $M$ and $N$ are ${\beta}_0(M) = \{(0,0)\}$ (green) and ${\beta}_k(M) = \emptyset$ for $k \geqslant 1$, and ${\beta}_0(N) = \{(\varepsilon,0),(0,\varepsilon)\}$ (green), ${\beta}_1(N) = \{(\varepsilon,\varepsilon)\}$ (red), and ${\beta}_k(N) = \emptyset$ for $k \geqslant 2$. Although $d_I(M,N) \leqslant \varepsilon$, there is no complete matching between the Betti numbers of $M$ and $N$. As a result, the bottleneck distance between, e.g., ${\beta}_0(M)$ and ${\beta}_0(N)$ is infinite, as any unmatched free summand is infinitely persistent.
• Figure 2: In botnan-lesnick, it is shown that a straightforward extension of the bottleneck distance to multiparameter interval decomposable modules (see \ref{['background-section']}) is not stable with respect to the interleaving distance. More specifically, it is shown that one can construct interval decomposable modules $M$ and $N$ such that $d_I(M,N)$ is arbitrarily small, and such that $M$ is indecomposable, $N$ decomposes into a direct sum of two indecomposable modules $N_1$ and $N_2$, and $d_I(M,N_1)$, $d_I(M,N_2)$, $d_I(N_1,0)$, and $d_I(N_2,0)$ are all large. Here, we illustrate a similar example, including the Betti signed barcodes $M$ and $N$. Even Betti numbers are shown in green and odd Betti numbers in red. As \ref{['main-theorem']} guarantees, there is a low-cost matching between the Betti numbers of $M$ and $N$, but, in order to construct this matching, the two Betti numbers of $M$ located at the corners of the thin region of its support must be matched together.
• Figure 3: The module $A_k$ of \ref{['possible-issue-example']}, when $k = 6$.
• Figure 4: The module $L_{a,b}$ of \ref{['instability-minimal-decomposition-example']} and the proof of \ref{['no-go-thm']}, when $n=2$.

Theorems & Definitions (57)

• Theorem 1.1
• Proposition 1.1
• Proposition 1.1
• Proposition 1.1
• Theorem 1.2
• Theorem 2.1: Hilbert
• Theorem 2.2: bjerkevik
• Lemma 3.1
• proof
• Proposition 3.1