Decomposing zero-dimensional persistent homology over rooted tree quivers

Luis Scoccola; Riju Bindua; Thomas Br\"ustle

REVIEW 2 major objections 6 minor 2 cited by

Decomposing zero-dimensional persistent homology over rooted tree quivers

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Pith's one-line read Tree-indexed zero-dimensional persistence is classifiable

desk verdict Genuine new finite-type result for zero-dimensional persistence over rooted tree posets, with a quadratic decomposition algorithm; the main claims hold, but two proof details need expansion. read the letter →

arxiv 2411.19319 v1 pith:3PZQ62KX submitted 2024-11-28 math.RT cs.CGmath.AT

Riju Bindua , Thomas Brüstle , Luis Scoccola This is my paper · ORCID

classification math.RTcs.CGmath.AT MSC 16G2055N31

keywords zero-dimensionalpersistenthomologyrootedtreequiversmodulesfiniterepresentationtypeelderrulemergetreesquiverrepresentationspersistence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Persistent homology is usually tractable because the indexing poset is a line, whose representations are of finite type. This paper shows the same tractability survives when the poset is a rooted tree: the objects that arise as zero-dimensional persistent homology, once closed under direct sums and direct summands, form a category of finite type. The indecomposable objects are exactly the reduced rooted tree modules, and there is a quadratic-time algorithm that decomposes any such representation. That makes tree-indexed clusterings, merge-tree morphisms, and restrictions of multiparameter filtrations amenable to the same decompose-into-features strategy that powers one-dimensional persistence.

What carries the argument

The load-bearing construction is the linearization of a rooted tree quiver over $Q$: a morphism $f: T \to Q$ is turned into a representation $k_T$ by pushing forward the constant representation of $T$, so each vertex of $T$ contributes a basis vector at its image in $Q$. The decomposition is controlled by the elder rule (Proposition 4.3): if two branches glued above the same vertex are comparable in the preorder on rooted tree quivers over $Q$, the smaller branch splits off as a direct summand. Reduced rooted tree quivers are those whose branches form antichains in this preorder; equivalently (Proposition 4.2) they are the ones admitting only the identity endomorphism, and their linearizations are the indecomposables. A gluing operation $G$ assembles rooted tree quivers by adjoining a new root, and representations and morphisms glue the same way.

What would settle it

Exhibit a finite rooted tree quiver $Q$ and a rooted tree module over $Q$ that is not isomorphic to a direct sum of reduced rooted tree modules, or an infinite family of pairwise non-isomorphic reduced rooted tree quivers over $Q$; either would directly contradict Corollary C.

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Extended reading notes

Core claim

Let $Q$ be a finite rooted tree quiver, the quiver analogue of a rooted tree poset. The paper establishes two characterisations. First, the representations obtainable as zero-dimensional persistent homology $\mathrm{H}_0$ of a $Q$-indexed filtration are precisely the finite direct sums of linearized rooted tree quivers over $Q$ (Theorem A(1)). Second, the additive closure of this class is precisely the category of finite direct sums of rooted tree modules over $Q$ (Theorem A(2)). The main structural result (Theorem B) says every rooted tree module over a rooted tree quiver splits as a direct sum of reduced rooted tree modules, which are indecomposable; hence the additive closure is of finite type and its indecomposables are these reduced modules (Corollary C). The proof runs through an elder rule for the preorder on rooted tree quivers over $Q$, and the same rule yields algorithms that decompose a linearized tree in $O(|T|^2)$ time and the zero-dimensional persistent homology of a $Q$-filtered graph in $O(|G|^2)$ time (Theorem D).

Load-bearing premise

The classification rests on the claim that, for a fixed finite rooted tree quiver, there are only finitely many reduced rooted tree quivers over it up to isomorphism; the paper states this follows by induction but leaves the induction implicit.

Editorial extensions

If this is right

Every zero-dimensional persistent homology module indexed by a rooted tree poset has a unique decomposition into reduced rooted tree modules, so the multiset of summands is a well-defined statistic of the filtration.
The decomposition of a linearized rooted tree and of the $\mathrm{H}_0$ of a filtered graph can be computed in quadratic time, making the classification usable in practice.
Each morphism between merge trees gives a representation of the target merge tree that decomposes by the same algorithm, providing an invariant of the morphism.
Restricting a multi-parameter filtration to any rooted tree subposet yields a $\mathrm{H}_0$ module that can be fully decomposed, turning a generally wild problem into a tractable one on the restriction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's statements, the multiset of reduced tree summands could serve as a feature vector for tree-indexed clusterings, since the decomposition is unique and computable in quadratic time.
The elder-rule mechanism suggests a template for other posets: whenever a preorder makes linearized branches form antichains after pruning, the same finite-type conclusion may hold.
Since the paper notes that higher-degree homology of a finite poset sees all representations, the finite-type phenomenon is specific to degree zero; this marks a boundary worth testing for other homology functors.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a real result, and the main theorems are correct. The paper shows that for a rooted tree quiver Q, the additive closure of the essential image of zero-dimensional persistent homology is of finite type, with indecomposables exactly the reduced rooted tree modules (Corollary C), and it gives a quadratic-time decomposition algorithm. That is genuinely new in persistence theory, where non-linearly indexed posets are usually wild. The algebraic characterization in Theorem A and the decomposition result in Theorem B are proved from scratch, not assumed; Kinser's work appears as definitional background and comparison, which is appropriate.

What the paper does well: the categorical setup is clean, the linearization functor is described inductively, and the elder rule (Prop 4.3) is proven with an explicit isomorphism. The algorithms come with correctness invariants and complexity bounds that look right. The merge-tree and restriction applications in Section 6 are sensible and modest.

Soft spots, in order of seriousness. First, the proof of (3)⇒(1) in Proposition 4.1 is not written correctly. The printed argument says that a nonzero root morphism gives, for each branch i of Q, some source branch j and target n with a nonzero root entry. But Definition 2.14 requires this for every source branch j. The missing step is compatibility along the arrow from the root of Q_i to the root of Q: the naturality square forces each row (or column, depending on convention) of the root block to sum to the nonzero global scalar, so every source branch has at least one nonzero entry. This is fixable, and the claim is true, but the proof as printed is incomplete. Because Prop 4.1 feeds into the elder rule and Theorem 4.6, it should be corrected.

Second, the finiteness argument in Corollary C is compressed into one sentence. The induction is routine and the claim is true, but given that finite type is the paper's headline conclusion, the referee should ask for the induction to be written out.

Third, Lemma 3.5 (the equivalence between set-valued functors and disjoint unions of rooted tree quivers over Q) is stated without proof. It is straightforward, but it supports Theorem A(1); a sketch would help. There are also a few minor typos.

Who should read it: persistence theorists, especially people working with merge trees or hierarchical clustering, and quiver representation people interested in tree posets. It deserves a serious referee and, after minor revision, acceptance. I would bring it to reading group.

Referee Report

2 major / 6 minor

Summary. This paper studies the linear representations of a rooted tree quiver Q obtained by applying zero-dimensional persistent homology to Q-indexed filtrations of topological spaces (equivalently, set-valued functors). The main results are: Theorem A characterizes the essential image repH0(Q) as the finite direct sums of linearized rooted tree quivers over Q, and its additive closure as the finite direct sums of rooted tree modules; Theorem B shows every rooted tree module over Q decomposes as a direct sum of reduced rooted tree modules; Corollary C concludes that add(repH0(Q)) is of finite type, with indecomposables precisely the reduced rooted tree modules; and Theorem D provides quadratic-time algorithms for the decomposition. The proofs are built on an inductive description of rooted tree quivers over Q, a preorder ≼_Q, and an explicit elder rule (Proposition 4.3).

Significance. If the results hold, the paper makes a valuable contribution to both representation theory and persistence theory: it identifies a natural subcategory of representations of rooted tree quivers—those arising from zero-dimensional persistent homology—that is of finite type even though the ambient category rep(Q) is generally wild, and it provides a concrete quadratic-time decomposition algorithm. The paper redevelops rather than black-boxes Kinser's theory, proves the elder rule by an explicit isomorphism, and gives correctness proofs by invariant for the algorithms. The finite-type classification and the algorithmic results are concrete and falsifiable, and the connection to merge-tree morphisms in Section 6 indicates useful applications. The main caveats are two proof gaps identified below, both repairable.

major comments (2)

[Proposition 4.1, proof of (3)⇒(1)] The proof as printed does not establish the implication: it shows only that for each branch i there exist j and n with φ_{i,j,n} nonzero at the root of Q_i, whereas Definition 2.14 requires, for every i and every j, the existence of some n with S^j_i ≼_{Q_i} T^n_i. The missing step is a column-sum argument at the edge σ_i→σ: since the structure maps from σ_i to σ are sums of identity maps, the compatibility condition forces every column of the root matrix of φ at σ_i to have sum equal to the nonzero root scalar λ, hence each column has a nonzero entry. With that argument, induction applies to every j. Because this implication is used in Proposition 4.3, Lemma 4.5, Theorem 4.6, and Theorem A(2), the proof must be corrected.
[Corollary C] The finite-type conclusion depends on the assertion that there are finitely many isomorphism classes of reduced rooted tree quivers over a fixed Q, but the proof is a single sentence referring to Definition 2.15 and induction. Please spell out the induction: for each vertex x of Q, the fiber of a reduced T at x is, for each child Q_i of x, an antichain in the finite poset of reduced rooted tree quivers over Q_i (finite by induction), and the height of any T is bounded by |Q|; this gives the required finiteness. Without this step the 'finite type' claim in Corollary C is unsupported.

minor comments (6)

[Lemma 2.3] The phrase 'join x∨y (i.e., greatest lower bound)' is incorrect: the join is the least upper bound; the greatest lower bound is the meet.
[Definition 2.18] In the second bullet, the lists are denoted N•_1,...,N•_n, but Q has k branches Q_1,...,Q_k; the index should be k.
[Proof of Theorem A(2)] The sentence 'where d∈N is such that, and note that, if succ^d(x) is the root of Q' is garbled; it should say 'where d is the unique integer such that succ^d(x) is the root of Q'.
[Proposition 5.1, correctness proof] In the paragraph checking condition (2), the sentence 'If x has no predecessors, then this tree quiver is the trivial rooted tree quiver, and condition (1) is met' appears to refer to condition (2); please correct the cross-reference.
[Proposition 5.1, invariant condition (3) proof] The reference to 'Definition 2.1' for the preorder ≼ should be to Definition 2.14.
[Algorithm 3, lines 14–16] The set T^{ℓ+1}_0 is used at ℓ=maxℓ, where it is not defined; clarify that it is empty in that case or adjust the loop bounds.

Circularity Check

0 steps flagged · score 0.0 of 10

The paper's central derivation is self-contained and does not reduce to its own inputs.

full rationale

I traced the main results back along the paper's own proofs. Theorem A(1) is proved from the equivalences between set-valued functors, disjoint unions of rooted tree quivers, and their linearizations (Lemmas 3.5, 3.6, 3.8), not assumed from prior work. Section 4 reproves the needed representation-theoretic facts: Proposition 4.1 gives an inductive proof characterizing the preorder, Proposition 4.3 proves the elder rule with an explicit isomorphism, and Theorem 4.6 proves indecomposability via local endomorphism rings. Theorem B, Corollary C, and Theorem A(2) then follow from these internal results together with the already-proved Theorem A(1). The finiteness step in Corollary C is compressed into one sentence, but it is a finite induction over the inductive definition of rooted tree quivers over Q, using the antichain condition in Definition 2.15; it does not presuppose the conclusion. The proof of Proposition 4.1(3)⇒(1) is terse and arguably under-justified, but under-justification is a proof gap, not circularity, and the cited Kinser results are external and independently formulated rather than being a self-citation chain. Self-citations such as [25] appear only as contextual references for clustering and elder-rule variants and are not load-bearing. No fitted parameter is renamed as a prediction, and no definition is constructed in terms of the target result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters, no fitted constants, and no new postulates. It relies on standard results in quiver representation theory and algebraic topology, and on the explicit finiteness of reduced rooted tree quivers, which is proven by induction. The field k is arbitrary and kept fixed throughout.

assumptions (6)

standard math Finite-dimensional representations of a finite quiver satisfy the Krull-Schmidt property (unique indecomposable decomposition).
Invoked in Section 2.3 and in Corollary C to conclude that every representation decomposes uniquely into indecomposables; standard result in representation theory of finite-dimensional algebras.
standard math H0(-;k) is naturally isomorphic to free composed with pi0 on the category of topological spaces with finitely many path components (Lemma 3.7).
Used in the proof of Theorem A(1) to identify the essential image of H0 with that of the free-vector-space functor on set-valued functors.
standard math Every functor Q -> set is isomorphic to pi0 of a functor Q -> top given by the discrete topology (Lemma 3.8).
Used in the proof of Theorem A(1) to show every set-valued functor is realized by zero-dimensional homology of a space-valued functor.
standard math The quiver representation category rep(Q) is equivalent to the category of functors from the path category of Q to vec (Section 2.3 and Definition 3.1).
Basis for translating between poset representations and quiver representations; used throughout the paper.
domain assumption Rooted tree quivers are finite, and all representations are finite-dimensional over a fixed field k.
Stated at the start of Section 2; the algorithms have complexity polynomial in the size of the rooted tree or graph, and all representation-theoretic finiteness statements depend on finite-dimensionality.
standard math An object with local endomorphism ring is indecomposable ([2, Corollary I.4.8(a)]).
Used in the proof of Theorem 4.6 to show reduced rooted tree modules are indecomposable.

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Pith. "Pith review of Decomposing zero-dimensional persistent homology over rooted tree quivers." pith.science (2026). https://pith.science/paper/3PZQ62KX

@misc{pith2026241119319,
  author       = {Pith},
  title        = {Pith review of: Decomposing zero-dimensional persistent homology over rooted tree quivers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3PZQ62KX}},
  note         = {Machine review of arXiv:2411.19319}
}

read the original abstract

Given a functor from any category into the category of topological spaces, one obtains a linear representation of the category by post-composing the given functor with a homology functor with field coefficients. This construction is fundamental in persistence theory, where it is known as persistent homology, and where the category is typically a poset. Persistence theory is particularly successful when the poset is a finite linearly ordered set, owing to the fact that in this case its category of representations is of finite type. We show that when the poset is a rooted tree poset (a poset with a maximum and whose Hasse diagram is a tree) the additive closure of the category of representations obtainable as zero-dimensional persistent homology is of finite type, and give a quadratic-time algorithm for decomposition into indecomposables. In doing this, we give an algebraic characterization of the additive closure in terms of Ringel's tree modules, and show that its indecomposable objects are the reduced representations of Kinser.

Figures

Figures reproduced from arXiv: 2411.19319 by the authors.

**Figure 1.** A rooted tree can be seen both as a poset and as a quiver. Left. The Hasse diagram of a rooted tree poset P. Right. The corresponding rooted tree quiver QP . The categories of representations are equivalent rep(P) ≃ rep(QP ) (Lemma 2.8). Zero-dimensional persistent homology. In several applications, notably clustering [8, 7, 25], the most relevant homological degree is zero. Since the zero-dimensional homology of a … view at source ↗

**Figure 2.** A rooted tree quiver T over a rooted tree quiver Q and its corresponding representation kT ∈ rep(Q). Left. An illustration of T −→ Q given by labeling the vertices of Q with distinct letters, and labeling the vertices of T with the label of their image. Center. Another illustration of T −→ Q, as well as its construction as an inductive rooted tree quiver over an inductive rooted tree quiver. Right. The linearization… view at source ↗

**Figure 3.** Left. Two filtrations f, g : K −→ P of a simplicial complex K (with vertices depicted as squares) by a linear ordered set P such that f ≤ g. Center. The connected components of the filtrations f and g as quivers over QP . Right. The decomposition of the homology H0(g) as a representation of the rooted tree quiver Σπ0(f) given by the connected components of f. See Section 6.1 for details. Proof. Thanks to Proposition… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Left. The Hasse diagram of a two-dimensional grid poset R (i.e., a product of two linear orders). Center and right. Restrictions of R that are rooted tree posets. assumption since one can work component-by-component, in the sense that X and Y decompose as a disjoint u…

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Claire Amiot, Thomas Br¨ ustle, and Eric J. Hanson. Invar iants of persistence modules deﬁned by order- embeddings, 2024

work page 2024

[2] [2]

Elements of the representation theory of associative algebras

Ibrahim Assem, Daniel Simson, and Andrzej Skowro´ nski. Elements of the representation theory of associative algebras. Vol. 1 , volume 65 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, 2006. Techniques of represen tation theory

work page 2006

[3] [3]

Botnan, Steﬀen Oppermann, and Jo han Steen

Ulrich Bauer, Magnus B. Botnan, Steﬀen Oppermann, and Jo han Steen. Cotorsion torsion triples and the representation theory of ﬁltered hierarchical cluster ing. Advances in Mathematics, 369:107171, 2020

work page 2020

[4] [4]

An introductio n to multiparameter persistence

Magnus Bakke Botnan and Michael Lesnick. An introductio n to multiparameter persistence. In Repre- sentations of algebras and related structures , EMS Ser. Congr. Rep., pages 77–150. EMS Press, Berlin, 2023. DECOMPOSING ZERO-DIMENSIONAL HOMOLOGY OVER ROOTED TREE QU IVERS 19

work page 2023

[5] [5]

O n the complexity of zero-dimensional multi- parameter persistence, 2020

Jacek Brodzki, Matthew Burﬁtt, and Mariam Pirashvili. O n the complexity of zero-dimensional multi- parameter persistence, 2020

work page 2020

[6] [6]

Coarse nodal count and topological persi stence

Lev Buhovsky, Jordan Payette, Iosif Polterovich, Leoni d Polterovich, Egor Shelukhin, and Vukaˇ sin Stojisavljevi´ c. Coarse nodal count and topological persi stence. J. Eur. Math. Soc. , 2024

work page 2024

[7] [7]

Eld er-rule-staircodes for augmented metric spaces

Chen Cai, W oojin Kim, Facundo M´ emoli, and Yusu W ang. Eld er-rule-staircodes for augmented metric spaces. SIAM J. Appl. Algebra Geom. , 5(3):417–454, 2021

work page 2021

[8] [8]

Guibas, Steve Y

Fr´ ed´ eric Chazal, Leonidas J. Guibas, Steve Y. Oudot, and Primoz Skraba. Persistence-based clustering in Riemannian manifolds. J. ACM , 60(6):Art. 41, 38, 2013

work page 2013