Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

TL;DR

The signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive.

Abstract

In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart in one-parameter persistence, the signed barcode decomposes the rank invariant as a $\Z$-linear combination of rank invariants of indicator modules supported on segments in the poset. We develop the theory behind these decompositions, both for the usual rank invariant and for its generalizations, showing under what conditions they exist and are unique. We also show that, like its unsigned counterpart, the signed barcode reflects in part the algebraic structure of the module: specifically, it derives from the terms in the minimal rank-exact resolution of the module, i.e., its minimal projective resolution relative to the class of short exact sequences on which the rank invariant is additive. To complete the picture, we show some experimental results that illustrate the contribution of the signed barcode in the exploration of multi-parameter persistence modules.

Figures (25)

• Figure 1: A one-parameter persistence module $M$ (top) indexed over $\{1, 2, 3, 4, 5\}\subset{\mathbb R}$, and the graphical representation of its barcode (in blue). The corresponding rank decomposition $\mathop{\mathrm{Rk}}\nolimits M = \mathop{\mathrm{Rk}}\nolimits \mathbf{k}_{\llbracket1,2\rrbracket} + \mathop{\mathrm{Rk}}\nolimits \mathbf{k}_{\llbracket2,5\rrbracket} + \mathop{\mathrm{Rk}}\nolimits \mathbf{k}_{\llbracket4,5\rrbracket}$ is readily available, and the ranks can easily be read from it: for instance, the rank $\mathop{\mathrm{Rk}}\nolimits M(2,4)=1$ is given by the one bar (thickened) that connects the down-set $2^-$ to the up-set $4^+$.
• Figure 2: The indecomposable module $M$ on the left-hand side does not have the same rank invariant as any direct sum of interval modules on the $3 \times 3$ grid. However, $\mathop{\mathrm{Rk}}\nolimits M$ is equal to the difference between the rank invariants of two direct sums of interval modules, as shown on the right-hand side. Blue is for intervals counted positively in the decomposition, while red is for intervals counted negatively.
• Figure 3: Minimal rank decomposition of the rank invariant of the module $M$ from Figure \ref{['fig:decomp_indec2_int_grid']} over the collection of segments (rectangles) in the $3\times 3$ grid. Blue is for rectangles in $\mathcal{R}\xspace$, while red is for rectangles in $\mathcal{S}\xspace$. This decomposition is unique as long as the intervals are constrained to be segments, otherwise Figure \ref{['fig:decomp_indec2_int_grid']} would give another valid minimal decomposition.
• Figure 4: Left: the signed barcode corresponding to the rank decomposition of Figure \ref{['fig:decomp_indec2_grid']}. Each bar is the diagonal with positive slope of one of the segments (rectangles) involved in the decomposition, with the same color code (blue for positive sign, red for negative sign). Right: computing $\mathop{\mathrm{Rk}}\nolimits M(s,t)$ for a pair of indices $s\leq t$, by counting with signed multiplicity the bars that connect the down-set $s^-$ to the up-set $t^+$ (here only the thickened bar does so).
• Figure 5: Minimal rank-exact resolution of a finitely presented interval representation $M$ of ${\mathbb R}^2$ (in gray). For clarity, the support of the interval is superimposed with the support each term in the resolution. By construction, the alternating sum of the rank invariants of these terms is equal to $\mathop{\mathrm{Rk}}\nolimits(M)$, which can be readily seen on the picture as the ranks sum up to $1$ within the superimposed areas while the dimensions sum up to $0$ elsewhere.

Theorems & Definitions (116)

• Theorem 1.1: botnan2020decompositionCrawley-Boevey2012
• Example 1.2
• Example 1.3
• Definition 1.4
• Definition 1.5
• Proposition 2.1
• proof
• Remark 2.2
• Corollary 2.3
• Proposition 2.4