Interval-Decomposable Modules

• Interval-decomposable modules are persistence modules over finite posets that decompose into interval modules supported on connected convex subsets.
• They leverage compression systems, Möbius inversion, and interval replacement to capture fine structural invariants and support efficient computational analysis.
• This framework is central to topological data analysis, providing a precise categorical method for approximating and computing persistence features.

An interval-decomposable module is a persistence module over a finite poset (or grid, or filtered category) that decomposes as a direct sum of interval modules, each supported on a connected convex subset of the index poset. Interval-decomposable approximations and the associated interval-rank invariants provide a categorical framework for analyzing and approximating persistence modules, with rich connections to algebraic invariants and computational algorithms. The following sections detail the key concepts, constructions, and formulas underpinning this theory, referencing developments such as compression systems, Möbius-inversion-based replacements, invariance properties, efficient computational frameworks, and the role of essential covering maps such as zigzag posets (Asashiba et al., 2024, Asashiba et al., 2024).

1. Compression Systems and Interval Modules

A compression system $\xi$ for a finite poset $P$ is a family of poset morphisms $\xi_I: Q_I\to P$ indexed by connected convex intervals $I\subseteq P$ (i.e., interval subposets). Each $Q_I$ is a finite connected poset, and $\xi_I$ factors the inclusion $I\hookrightarrow P$, ensuring $\xi_I(Q_I)\supseteq I$, with images containing minimal and maximal elements of $I$, and paths in $Q_I$ realizing every segment $[x,y]\subseteq I$. Examples include:

• Total compression ($\xi$ assigns each $I$ its inclusion into $P$): captures classical restriction to interval subposets.
• Source-sink compression (restricting to minimal/maximal points of $I$): tracks data only at endpoints.

Each $\xi$ induces a restriction functor $R_I = F_I^*$ from modules over $P$ to $Q_I$, and the interval module $V_{Q_I}$ is the unique indecomposable in $\operatorname{mod} Q_I$ supported on $Q_I$.

2. The Interval-Rank Invariant and Möbius Inversion

Given $M\in \operatorname{Mod} k[P]$ and a compression system $\xi$, the $I$-rank invariant is the multiplicity $c_I(M)$ of $V_{Q_I}$ in the Krull-Schmidt decomposition of the restricted module $R_I(M)$:

$$\mathrm{rank}_I^\xi(M) := c_I(M) \in \mathbb{Z}_{\geq 0}.$$

The family $\{\mathrm{rank}_I^\xi(M)\}_{I\in \mathcal I}$ generalizes the rank invariant studied by Kim–Mémoli, which coincides with total compression. These ranks encode fine structural information about $M$ and are crucial for both theoretical and computational analyses.

To recover the interval-decomposable approximation, signed interval multiplicities $d_I^\xi(M)$ are computed via Möbius inversion on the poset of intervals $(\mathcal I,\subseteq)$:

$$d_I^\xi(M) = \sum_{I \leq J} \mu(I, J) \, \mathrm{rank}_J^\xi(M),$$

where $\mu$ is the Möbius function.

3. Interval Replacement and Its Properties

The interval replacement $\operatorname{IntRep}_\xi(M)$ is the formal sum in the split Grothendieck group $$K_0^{\operatorname{split}}(\operatorname{Mod} k[P])$$: $$\operatorname{IntRep}_\xi(M) = \sum_{d_I^\xi(M) > 0} d_I^\xi(M)[V_I] - \sum_{d_I^\xi(M) < 0} (-d_I^\xi(M))[V_I].$$ This yields a pair of genuine interval-decomposable modules—the positive and negative parts—whose difference in $K_0^\operatorname{split}$ provides a best interval-decomposable approximation to $M$ in the sense of preserving interval-rank invariants. In the case that $M$ is genuinely interval-decomposable, $\operatorname{IntRep}_\xi(M)$ coincides with $M$.

Table: Interval Decomposition Steps

Step	Operation / Output	Key Concept
Compression system $\xi$	Associated $Q_I, \xi_I$ for each $I$	Functorial restriction
Restriction	$R_I(M)$ over $Q_I$	Localization
Interval-rank	$c_I(M)$	Krull-Schmidt multiplicity
Möbius inversion	$d_I^\xi(M)$	Signed barcodes
Interval replacement	$\operatorname{IntRep}_\xi(M)$	Split Grothendieck approximation

4. Invariance and Explicit Formulas

The formation of $\operatorname{IntRep}_\xi(M)$ preserves the interval-rank invariant: for any interval $I$, $$\mathrm{rank}_I^\xi(\operatorname{IntRep}_\xi(M)) = \mathrm{rank}_I^\xi(M)$$ (Asashiba et al., 2024). This preservation is stronger than simply retaining the standard rank invariant and ensures that dimension vectors and higher-order invariants such as rank functions for all pairs are matched.

For a given compression system, $\mathrm{rank}_I^\xi(M)$ can be computed explicitly: $$\mathrm{rank}_I^\xi(M) = \operatorname{rank}([M(\xi_I(b_1)\to \xi_I(a_1)) \ldots ]) - \operatorname{rank}(M_1) - \operatorname{rank}(M_2)$$ where $M_1, M_2$ are concrete block matrices built from structure maps between sources and sinks, as in Theorem 5.23 of (Asashiba et al., 2024). Thus, interval-rank invariants are amenable to direct linear algebraic calculation using only the underlying module data.

5. Essential Coverings and Recovery of Total Rank

A compression system $\xi_I$ essentially covers $I$ if every term required for the total-rank formula (e.g., Kim–Mémoli’s rank) can be realized by structure maps coming from $Q_I$ via $\xi_I$. If this essential covering holds, then for all $M$: $$\mathrm{rank}_I^\xi(M) = \mathrm{rank}_I^{\operatorname{tot}}(M)$$ and the interval-rank invariant under $\xi$ recovers the original total rank invariant. Zigzag posets or Dynkin type $\mathbb{A}$ covers—where intervals are traversed with paths in a chain—realize this situation, enabling highly efficient computations and direct application to classical TDA workflows.

A worked example on a 3-element chain demonstrates the computation and Möbius inversion that realizes the interval replacement, illustrating coincidence of dimension vectors and ranks with the maximal interval-decomposable summand.

6. Computational Implications and Complexity

The explicit formulas for interval multiplicities $d_M(V_I)$ (see (Asashiba et al., 2024)) further allow computation of the maximal interval-decomposable summand of an arbitrary module by evaluating the rank of carefully constructed matrices from the module’s structure maps. In standard scenarios, such as when the covering poset $Z$ is Dynkin type $\mathbb{A}$ (zigzags), the calculation is especially efficient. The necessary matrices are small, the enumeration of intervals is quadratic in the poset size, and well-optimized implementations avoid the need to compute the full module structure maps.

Practical implications include:

• Feasibility for moderate-size posets.
• Output-sensitive computational complexity in rank computations.
• Direct applicability to real data via zigzag chain reductions.

7. Theoretical and Algorithmic Significance

Interval replacements extend the applicability of interval-decomposability theory from classical (totally ordered or grid) settings to arbitrary finite posets, parametrized by the choice of compression system. Essential covers link efficient matrix-based algorithms to the underlying algebraic invariants of persistence modules, ensuring that interval-decomposable approximations capture all interval-rank invariants and are compatible with standard approaches when applicable.

The quantification and preservation of interval-rank invariants position interval replacements as central tools in both the theoretical study and practical computation of barcodes and persistence features in algebraic topological data analysis (Asashiba et al., 2024, Asashiba et al., 2024).