Signed Rectangle Rank in Multiparameter Persistence

• Signed rectangle rank is an integer-valued function that decomposes the rank invariant of a multiparameter persistence module into signed contributions from axis‐aligned rectangles.
• It employs Möbius inversion on the poset of rectangles to compute discrete mixed derivatives, providing a canonical, yet sometimes unstable, barcode representation.
• Recent comparisons with hook-based resolutions illustrate the trade-offs between geometric intuition and bottleneck stability in higher-dimensional persistence analysis.

Searching arXiv for the core persistence papers and related work on signed rectangle rank. Signed rectangle rank is an integer-valued coefficient function that appears in rank decompositions of multi-parameter persistence modules. For a persistence module $M$ indexed by a poset such as a finite grid or $\mathbb{R}^n$, it records how the rank invariant $r_M(u \le v)=\operatorname{rank}(M(u\to v))$ can be reconstructed as a signed $\mathbb{Z}$-linear combination of rank invariants of indicator modules supported on axis-aligned rectangles (Botnan et al., 2021). In this sense, signed rectangle rank generalizes the unsigned interval multiplicities of one-parameter barcodes to the multi-parameter setting, where cancellations are generally unavoidable. The notion is closely tied both to Möbius inversion on the poset of rectangles and to the homological structure of the module through rank-exact resolutions (Botnan et al., 2021). Subsequent work distinguishes the canonical minimal rectangle decomposition from a different homological signed decomposition by hooks, showing that the former is not bottleneck stable in dimension at least two, whereas the latter is stable (Botnan et al., 2022).

1. Definition and ambient setting

Let $P$ be a poset viewed as a category, and let $M:P\to \mathrm{Vec}_k$ be a pointwise finite-dimensional persistence module. The rank invariant is the function

$r_M(u\le v):=\operatorname{rank}(M(u\to v)),$

defined on

$$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$

(Botnan et al., 2021).

A closed segment in $P$ is the down-up interval

$\{u,v\}:=\{x\in P\mid u\le x\le v\}.$

In multi-parameter settings such as $\mathbb{R}^n$0 with product order, a basic class of such segments is given by axis-aligned rectangles

$\mathbb{R}^n$1

(Botnan et al., 2021).

For any segment or interval $\mathbb{R}^n$2, the indicator module $\mathbb{R}^n$3 assigns $\mathbb{R}^n$4 on $\mathbb{R}^n$5 and $\mathbb{R}^n$6 outside, with identity structure maps inside $\mathbb{R}^n$7 and zero maps otherwise. Its rank invariant satisfies

$\mathbb{R}^n$8

(Botnan et al., 2021).

With a chosen family $\mathbb{R}^n$9 of rectangles, a signed rectangle rank decomposition of $r_M(u \le v)=\operatorname{rank}(M(u\to v))$0 is an identity

$r_M(u \le v)=\operatorname{rank}(M(u\to v))$1

with pointwise finite sum on $r_M(u \le v)=\operatorname{rank}(M(u\to v))$2 (Botnan et al., 2021). The resulting map $r_M(u \le v)=\operatorname{rank}(M(u\to v))$3 is the signed rectangle rank of $r_M(u \le v)=\operatorname{rank}(M(u\to v))$4 relative to $r_M(u \le v)=\operatorname{rank}(M(u\to v))$5. Equivalently, one may package positive and negative coefficients as a signed barcode

$r_M(u \le v)=\operatorname{rank}(M(u\to v))$6

where $r_M(u \le v)=\operatorname{rank}(M(u\to v))$7 and $r_M(u \le v)=\operatorname{rank}(M(u\to v))$8 are direct sums of rectangle modules determined by the positive and negative multiplicities (Botnan et al., 2021).

In the language of later work, this is the signed difference of rectangle-module rank contributions in the unique minimal rectangle decomposition $r_M(u \le v)=\operatorname{rank}(M(u\to v))$9 for finitely presented $\mathbb{Z}$0-modules (Botnan et al., 2022).

2. Relation to barcodes and multi-parameter persistence

The conceptual role of signed rectangle rank is to provide a multi-parameter analogue of the one-parameter persistence barcode. In one parameter, the rank invariant admits an unsigned decomposition into interval ranks, so the multiplicities are nonnegative. In multiple parameters, the rank invariant cannot generally be written as an unsigned sum of rectangle indicator ranks; signed coefficients are needed to account for overlaps and algebraic dependencies (Botnan et al., 2021).

The associated signed barcode represents each rectangle geometrically by its main diagonal bar, with multiplicity and sign. Positive and negative rectangles encode contributions and cancellations whose cumulative effect reconstructs the full rank invariant (Botnan et al., 2021). This makes signed rectangle rank a compressed global encoding of the rank invariant, rather than merely a local statistic.

The paper introducing signed barcodes develops the notion not only for ordinary segment ranks but also for generalized ranks over intervals,

$\mathbb{Z}$1

with the usual rank recovered when $\mathbb{Z}$2 is a closed segment (Botnan et al., 2021). This broader framework places signed rectangle rank within a family of decompositions of generalized rank functions over interval dictionaries.

A recurrent source of ambiguity in the literature is the phrase “signed rectangle rank,” which also appears in a different field: communication complexity. There it denotes, for a Boolean matrix $\mathbb{Z}$3, the minimum number of all-1 rectangles needed to express $\mathbb{Z}$4 as a $\mathbb{Z}$5-sum, and it satisfies

$\mathbb{Z}$6

with $\mathbb{Z}$7 for matrices of rank $\mathbb{Z}$8 (Hambardzumyan et al., 2 Oct 2025). The persistence-theoretic notion is distinct: it is not a complexity measure of a matrix, but an integer coefficient function in a decomposition of the rank invariant of a persistence module (Botnan et al., 2021). The shared terminology reflects the same combinatorial idea of reconstructing an object by signed rectangle contributions, but the ambient categories, invariants, and goals are different.

3. Existence, uniqueness, and canonical minimality

The foundational structural result is a basis theorem for rank functions. Let $\mathbb{Z}$9 be a locally finite collection of intervals in a poset $P$0, and let $P$1 have locally finite support. Then

$P$2

uniquely, with pointwise finite sum (Botnan et al., 2021). In particular, the family $P$3 acts as a basis of rank functions with locally finite support.

Specialized to rectangles, this yields existence and uniqueness of signed rectangle decompositions under natural finiteness assumptions. The corresponding minimal signed decomposition, expressed as disjoint multisets $P$4, is unique (Botnan et al., 2021). On finite posets, and in particular on finite grids $P$5, every module rank invariant admits such a unique minimal signed decomposition over grid rectangles (Botnan et al., 2021). For finitely presented modules over $P$6, the usual rank admits unique minimal rank decompositions over right-open rectangles (Botnan et al., 2021).

Later work reformulates this as a canonical signed barcode. For any finitely presented $P$7, there exists a unique signed $P$8-barcode

$P$9

such that the modules appearing are right open rectangle modules, the positive and negative multisets are disjoint, and the signed barcode is a rank decomposition of $M:P\to \mathrm{Vec}_k$0 (Botnan et al., 2022). This formalizes signed rectangle rank as a canonical object rather than an arbitrary choice of decomposition.

Even without local finiteness, uniqueness persists at the minimal level whenever a decomposition exists: the minimal decomposition is obtained by cancelling common intervals from any decomposition, and any two decompositions satisfy a balancing identity

$M:P\to \mathrm{Vec}_k$1

(Botnan et al., 2021). This gives signed rectangle rank a canonical status whenever the relevant decomposition class is fixed.

4. Möbius inversion and explicit coefficient formulas

A central computational fact is that the coefficients $M:P\to \mathrm{Vec}_k$2 are obtained by Möbius inversion on the poset of rectangles ordered by inclusion. If $M:P\to \mathrm{Vec}_k$3 denotes the Möbius inverse of the zeta function on that poset, then for a rank function $M:P\to \mathrm{Vec}_k$4,

$M:P\to \mathrm{Vec}_k$5

for every interval $M:P\to \mathrm{Vec}_k$6 (Botnan et al., 2021). Thus the minimal decomposition is realized directly by the integer-valued function $M:P\to \mathrm{Vec}_k$7, split into positive and negative parts.

On a finite grid $M:P\to \mathrm{Vec}_k$8, for a rectangle $M:P\to \mathrm{Vec}_k$9 with $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$0, the coefficient is the mixed finite difference

$r_M(u\le v):=\operatorname{rank}(M(u\to v)),$1

(Botnan et al., 2021). Equivalently, using backward differences at lower corners and forward differences at upper corners,

$r_M(u\le v):=\operatorname{rank}(M(u\to v)),$2

(Botnan et al., 2021). Expanded over $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$3,

$r_M(u\le v):=\operatorname{rank}(M(u\to v)),$4

with the convention that values outside the grid are zero (Botnan et al., 2021).

These identities show that signed rectangle rank is a discrete mixed derivative of the rank invariant. The original paper describes a heuristic continuous interpretation for finitely presented $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$5-modules: $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$6 behaves like a signed “jump measure” of $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$7 across lower and upper faces of rectangles (Botnan et al., 2021). This suggests that signed rectangle rank localizes the singular support of the rank invariant in a combinatorial form.

The computational consequence on finite grids is explicit. If rank queries $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$8 are available for all comparable pairs, all rectangle coefficients can be computed by the $r_M(u\le v):=\operatorname{rank}(M(u\to v)),$9-fold finite-difference formula, after which positive coefficients populate $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$0 and negative coefficients populate $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$1 (Botnan et al., 2021). The arithmetic complexity is

$$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$2

which is linear in the encoding size of $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$3 for fixed $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$4 (Botnan et al., 2021).

5. Algebraic origin in rank-exact resolutions

Signed rectangle rank is not merely a combinatorial inversion artifact; it reflects the algebraic structure of the module. The key notion is that of a rank-exact short exact sequence

$$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$5

meaning that the generalized rank is additive: $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$6 pointwise on segments (Botnan et al., 2021). This class of sequences defines an exact structure $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$7 on the category of persistence modules (Botnan et al., 2021).

Within this exact category, the Grothendieck group $$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$8 is generated by classes of indicator modules of segments, and for finite posets the rank invariant induces an isomorphism

$$\mathrm{Relation}(P):=\{(u,v)\in P\times P\mid u\le v\}$$9

with basis $P$0 (Botnan et al., 2021). If a module $P$1 admits a finite rank-exact projective resolution

$P$2

then in $P$3,

$P$4

hence

$P$5

(Botnan et al., 2021). Since each $P$6 decomposes as a direct sum of indicator modules on segments or rectangles, the coefficients in the signed rank decomposition arise as the alternating signs of the resolution terms (Botnan et al., 2021).

This homological origin becomes central in the stability theory developed later. In the rank exact structure on finitely presentable $P$7-modules, the indecomposable rank projectives are hook modules rather than rectangle modules (Botnan et al., 2022). Every such module admits a finite minimal rank projective resolution, and the associated signed barcode

$P$8

is well defined and reconstructs the rank invariant as a signed sum of hook-module rank invariants (Botnan et al., 2022). Thus the rectangle-based signed decomposition and the rank-exact homological decomposition are related but not identical. A plausible implication is that the minimal rectangle decomposition captures a canonical Möbius-theoretic compression of the rank invariant, whereas the rank-exact decomposition is the one directly aligned with projective homological structure.

6. Stability, limitations, and later refinements

A major issue for applications is stability under perturbations. The initial work reports that signed decompositions are “maximally separated among all decompositions in the matching distance on rank invariants” and that replacing a module by a module built from the positive and negative rectangle summands does not increase inter-module distances (Botnan et al., 2021). However, subsequent analysis identifies a stronger obstruction: the minimal rank decomposition by rectangles is not bottleneck stable in the natural signed bottleneck metric when $P$9 (Botnan et al., 2022).

The negative result is explicit. There is no function $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$0 with $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$1 as $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$2 such that

$\{u,v\}:=\{x\in P\mid u\le x\le v\}.$3

for all finitely presented $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$4 when $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$5 (Botnan et al., 2022). The counterexample uses right open rectangles $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$6, $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$7, and $\{u,v\}:=\{x\in P\mid u\le x\le v\}.$8, with

$\{u,v\}:=\{x\in P\mid u\le x\le v\}.$9

and constructs pairs of modules whose interleaving distance tends to zero while the signed bottleneck cost remains at least $\mathbb{R}^n$00 (Botnan et al., 2022). This shows that small perturbations can force large changes in the minimal signed rectangle barcode.

The remedy proposed in the same paper is to replace rectangle decompositions by the rank exact decomposition built from hook modules. For finitely presented modules $\mathbb{R}^n$01,

$\mathbb{R}^n$02

(Botnan et al., 2022). This signed hook barcode is therefore bottleneck stable with an explicit constant. The paper further proves universality properties for the signed bottleneck dissimilarity on hook barcodes and computes the global dimension of the rank exact structure as

$\mathbb{R}^n$03

(Botnan et al., 2022).

The main interpretive consequence is not that signed rectangle rank is invalid, but that it serves a different purpose. It remains a canonical and geometrically intuitive decomposition of the rank invariant, especially useful for visualization and algebraic inspection (Botnan et al., 2021). Yet if one requires perturbation-stable signed summaries, the hook-based rank exact decomposition is the preferred replacement (Botnan et al., 2022).

A related caveat concerns the choice of decomposition dictionary. Over all intervals rather than rectangles, decompositions may exist but need not be unique, and the resulting barcodes can be harder to interpret (Botnan et al., 2021). Restricting to rectangles on grids restores uniqueness and geometric transparency, but not bottleneck stability (Botnan et al., 2021, Botnan et al., 2022).

7. Interpretation, examples, and broader significance

In the simplest case, if the rank invariant is already the rank of a single rectangle module $\mathbb{R}^n$04, then the signed rectangle rank assigns coefficient $\mathbb{R}^n$05 to $\mathbb{R}^n$06 and $\mathbb{R}^n$07 elsewhere (Botnan et al., 2021). More interestingly, when the same region of the rank invariant can be generated by overlapping rectangles from incomparable lower corners, Möbius inversion returns positive coefficients on those birth rectangles together with a negative rectangle correcting the overlap (Botnan et al., 2021). This is the canonical multi-parameter counterpart of inclusion–exclusion.

In one parameter, no negative coefficients are needed: every coefficient $\mathbb{R}^n$08 is nonnegative, and the decomposition reduces to the usual barcode (Botnan et al., 2021). In multiple parameters, negative coefficients encode algebraic relations beyond simple feature counts. The signed barcode may therefore be read as a balance of positive and negative bars crossing parameter regions, with $\mathbb{R}^n$09 equal to the number of positive bars connecting $\mathbb{R}^n$10 to $\mathbb{R}^n$11 minus the number of negative ones (Botnan et al., 2021).

For finitely presented $\mathbb{R}^n$12-modules, computation proceeds by discretizing to the finite critical grid induced by generators and relations, then applying the same Möbius inversion formulas (Botnan et al., 2021). The paper also notes that smoothing commutes with signed decompositions and removes bars whose signed prominence lies near the diagonal hyperplanes, suggesting a denoising mechanism (Botnan et al., 2021). This suggests that signed rectangle rank is particularly suited to exploratory analysis of multi-parameter persistence, where one seeks a compact visual and algebraic summary of the global rank structure.

Within topological data analysis, the broader significance of signed rectangle rank lies in three facts. First, it gives a canonical decomposition of the rank invariant for finite grids and finitely presented modules (Botnan et al., 2021). Second, it exposes a direct bridge between combinatorial inversion formulas and homological algebra through rank-exact resolutions (Botnan et al., 2021). Third, the later instability results clarify its scope: it is best understood as a canonical signed encoding of rank information, not as the final stable metric invariant in higher-parameter settings (Botnan et al., 2022).

A common misconception is to treat the signed rectangle barcode as a straightforward multi-parameter replacement for the classical barcode in all respects. The later theory shows that this is too strong: it generalizes the decomposition aspect of one-parameter persistence, but not the bottleneck stability phenomenon associated with one-parameter barcodes (Botnan et al., 2022). The stable homological replacement uses hooks rather than rectangles, even though the latter remain more immediately geometric.