MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology
Authors: Juni Schindler
Mauricio Barahona
Venue: ICLR 2026 Poster (2026)
Keywords: topological data analysis
multiparameter persistent homology
multiscale clustering
non-hierarchical clustering
Sankey diagrams
Primary area: learning on graphs and other geometries & topologies

Abstract: Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness.  Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF is a complete invariant of (non-hierarchical) sequences of partitions and can be interpreted as a higher-order extension of Sankey diagrams, which reduce to dendrograms for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We then demonstrate through experiments the use of MCbiF Hilbert functions as interpretable topological feature maps for downstream machine learning tasks, and show that MCbiF feature maps outperform both baseline features and representation learning methods on regression and classification tasks for non-hierarchical sequences of partitions. We also showcase an application of MCbiF to real-world data of non-hierarchical wild mice social grouping patterns across time.

Published as a conference paper at ICLR 2026
MCBIF: MEASURING TOPOLOGICAL AUTOCOR- RELATION IN MULTISCALE CLUSTERINGS VIA 2-
PARAMETER PERSISTENT HOMOLOGY
Juni Schindler
Department of Mathematics, Imperial College London, UK
Department of Mathematical Modeling and Machine Learning, University of Zurich, Switzerland
Juni schi aadlerr@zz.cc
Mauricio Barahona
Department of Mathematics, Imperial College London, UK m. barahonaa imperi ial. all.ac...... ac.uk
ABSTRACT
Datasets often possess intrinsic multiscale structure with meaningful descrip- tions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., necessarily hierarchical sequences of par- titions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF is complete invariant of honn-hierarchica) sequences of partitions and can be interpreted as higher-order extension of Sankey diagrams, which reduce to dendrograms for hicrarchical se- quences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the re- finement order of partitions, whereas dimension one, the MPH captures higher- order inconsistencies between clusters across scales. We then demonstrate through experiments the use of MCbiF Hilbert functions as interpretable topological fea- ture maps for downstream machine learning tasks, and show that MCbiF feature maps outperform both baseline features and representation learning methods on regression and classification tasks for non-hierarchical sequences of partitions. We also showcase an application MCbiF to real-world data of non-hierarchical wild mice social grouping patterns across time.
INTRODUCTION
In many applications, datasets possess intrinsic multiscale structure, whereby meaningful descrip- tions exist at different resolutions or levels of coarseness. Think, for instance, of the multi-resolution
structure in commuter mobility patterns (Alessandretti al. 2020 Schindler et al.) 2023). commu- nities in social networks (Beguerisse-Daaz et al.) (017) and thematic groups of documents (Blei)
al. 2003 Grootendorst, 2022): the subgroupings in single-cell data (Hoekzema et al. 2022) or phylogenetic trees (Chan et al.][2013) and the functional substructures in proteins (Delvenne et al., 2010 Delmotte al.] 2011). In such cases, the natural description the data goes beyond a single clustering and consists of multiscale sequence of partitions parametrised by scale parameter t. Traditionally, multiscale descriptions have emerged from hierarchical clustering, with as the depth of the dendrogram (Carlsson & Mémoli 2010 Murtagh & Contreras) 2012). However, in many real- world applications, the data structure is multiscale yet non him-hhirrrchial For example in temporal clustering, where corresponds physical time (Rosvall & Bergstrom 2010 Liechti & Bonhoeff feer 2020 Bovet et al.) 2022);; in topic modelling and document classification, where captures the
coarseness of the topic groupings (Altuncu et al. 2019] Fukuyama al. 2023
Liu al. 2025): or

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in clustering methods that exploit a diffusion the data geometry, where t is the increasing time horizon of the diffusion (Coifman et al. 2005 Azran & Ghahramani 2006 Lambiotte et al. 2014).
A natural problem is then how to analyse and compare non-hierarchical. multi-resolation sequences of partitions parametrised by the scale t. Here we address this question from the perspective of topological data analysis (Carlsson & Zomorodian 2009) Carlsson et al. 2009; Botnan & Lesnick 2023) by introducing the Multiscale Clustering Bifiltration (MCbiF), 2- 2-parameter filtration of abstract simplicial complexes that encodes the patterns of cluster intersections across all scales.
Problem definition. A partition of set X {x1, 22,,..., 2N) is collection of mutually exclu- sive subsets C₁ C X (or clusters) that cover X, i.e,, π = {C1,.... Cc} such that X = U C and Cj = 0, Vi + j. We denote the cardinality as |T| = C, and n; for the i-th cluster C. of n. Let 1Ix denote the space of partitions of X. We write π S n if every cluster in π is contained in cluster of n². This refinement relation constitutes partial order and leads to the partition lattice
(Tx, S) with lower bound 0 {(z))....; {arw) } and upper bound i {X) (Birkhoff 1967).
Here, we consider a sequence of partitions defined as 0 [t1, 00) - IIx, t + 8(t) E I i.e., piccewise-constant function that assigns partition of X to each The scale has M change points 4 < l2 < < tM. so that O()) = ((4m) for t E (m,.4441) 1). m = 1,..., M - 1, and 0(t) B(tM) for t E [tas, 00). The sequence is hierarchical in [s, if we have a strict sequence of refinements: either agglomerative (0(r1) I 0(r2), Vri, E [s. f] with S r2) or divisive (0(r)) 2 (r22), Vri.12 E [s, with ri S r2). We say that strictly hierarchical it is hierarchical in (0). The sequence I is called goarse-graining if [0(s) 2 10(1), Vs S Conversely, 0 is called fine- -e-iiiiiinn if [0()) I (@(t)), Vs St.. t.
Our goal is to characterise and analyse sequences of partitions 9, including non- non-hierarchical ones, in an integrated manner, taking account of memory effects across the scale L.
Remark 1. Here, we are not concerned with the task of computing the multiscale clustering (i.e., the sequence of partitions 0) from dataset for which severul methods exist. Rather, take 0 as a given, and aim to analyse its structure.
This problem is distinct from consensus clustering, which aims to produce a summary partition by combining a set of partitions obtained, e.g., from different optimisations or clustering algorithms (Strell & Ghosh 2002 Vega-Pont Pons & Ruiz- Mhiz--hulcloper 2017).
Remark 2.
(a)
Imput: partitons
(b)
HFols g 0 112
((0)
8(1)
6(2)
((3)
2
Output: Multiccate Gustenigg Uliiiaaioon
KM
Kia
K
K
1
3
K
K
Kla
K
HF.(s.p)
K38
K3
KM
KM
2
K
Figure 1: (a) The MCbiF encodes the structure of a non-hierarchical sequence of partitions & as a bifiltration of abstract simplicial complexes Ke,t (see Example (37. (b) The Hilbert functions HF f) of the MCbiF are invariants that capture the topological autocorrelation of 0: violations of the refinement order at dimension k = O, and higher-order cluster inconsistencies at dimension k 1. The (F, t) can be used feature maps for downstream machine learning tasks.
'Coasse-raining is cquivalent non-decreasing mean cluster size (see Remark 23 in Appendix B.I
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Contributions. To address this problem, we define the MCbiF, bifiltration of abstract simpli- cial complexes, which represents the clusters and their intersection patterns in the sequence 0 for varying starting scale and lag t --s (Fig. D The MCbiF is complete invariant of 0, and use the machinery of multiparameter persistent homology (MPH) (Carlsson & Zoooorodian, 2009. Carlsson et al. 2009; Botnan & Lesnick 2023) to summarise its topological structure. We prove that the MCbiF leads to a block decomposable persistence module with stable Hilbert functions HF t), and we show that these invariants characterise the topological autocorrelation of the se- quence of partitions 0 across scales. In particular, the HF, (s, t) quantify the non- non-hierarchy in 0 in complementary ways: at dimension & = 0, detects the lack of a maximal partition in the subposet (((,,)) with respect to refinement; at dimension k 1, it quantifies higher-order inconsistencies of cluster assignments across scales. contrast, baseline methods such as ultrametries (Carlsson & Mémoli 2010) or information-based measures (Meilä) 2003) are restricted to pairwise comparisons between elements or clusters, respectively; hence, they cannot detect higher-order cluster inconsis- tencies. Furthermore, we provide an equivalent nerve-based construction of the MCbiF that can be interpreted as a higher-order extension of the Sankey diagram of the sequence of partitions. In par- ticular, for the hierarchical case, the 0 o-dimensional MCbiF Hilbert function can be obtained from the number of branches in the Sankey diagram, which reduces to dendrogram. Finally, we show that HF 1) provide interpretable feature maps usable for downstream machine learning tasks. In our experiments, the MCbiF feature maps outperform both baseline features and representation learning methods on regression and classification tasks for non- -on-hierarchical sequences of partitions. We also showcase an application of MCbiF to real-world data of non non-hienan -on-hicrarchiaa wild mice social grouping patterns across time (Bovet al. 2022).
2 RELATED WORK
Dendrograms and Ultrametries. A hierarchical, coarse-graining sequence 8 with 6(t1 0) ò and (((M) 1 is called agglomerative dendrogram, and can be represented by an acyclic rooted merge tree (Jain et al., 1999. Carlsson & Mémoli, (010). One can define an ultrametric Dg from the first-merge times, which follows from the depth in the dendrogram. Carlsson & Mémoli (2010) showed there is a one-to-one correspondence between agglomerative dendrograms and ultrametrics distance between the ultrametric spaces (Mémoli et al., 2023). When 0 is hirrheeral,,, however, first-merge times no longer define the sequence uniquely, as merged clusters can split again. Hence, 0 cannot be represented by a tree, and Dg does not fulfil the triangle inequality in general. As result, ultrametrics cannot be used to analyse and compare non-hierarchical sequences of partitions (see Section
that can be used to efficiently compare dendrograms using the
Pairwise Comparison of Partitions. Different measures to compare pairs of partitions have been introduced in the literature. The Adjusted Rand Index (ARI) is the chance-corrected Rand Index that compares clusterings by counting elements that are assigned to the same or different clusters (Hu- bert & Arabie 1985). Information-based measures compute the information gain and loss between two partitions using the Conditional Entropy (CE) or the Variation of Information (VI), which is metric on IIx (Meilā 2003; 2007) (see formulas in Appendix F). The Maximum Overlap Distance (MOD), which also metric on IIx. measures the minimal classification error when partition is assumed to be the correct one (Peixoto) 202I). A key limitation of all these measures is that they rely only on pairs of partitions and cannot capture higher-order cluster inconsistencies; yet tending them to more than two partitions is non-trivial. In consensus clustering, the average of the pairwise ARI, VI or MOD between the partitions in a set used as a consensus index (Vinh & Epps, 2009 Vinh et al. 2010). However, these average measures are insensitive to ordering and so cannot capture memory effects in sequences of partitions.
THE MULTISCALE CLUSTERING BIFILTRATION (MCBIF)
The central object of our paper is novel bifiltration of abstract simplicial complexes that encodes cluster intersection patterns in the sequence of partitions across the scale t. For background on simplicial complexes and bifiltrations, Appendix F.2
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Definition 3 (Multiscale Clustering Bifiltration), Given a sequence 0: (0) - IIx, define the Multiscale Clustering Bifiltration (MCbiF), M, as a bifiltration of abstract simplicial complexes:
M = (K* Sisse where K =
U U AC
(1)
1 SaSeer Ce0(r)
In this construction, each cluster C corresponds to ((CI-I))dimensionn)) solid simplex AC 2c which, by definition, contains all its lower-dimensional simplices (Schindler & Barahona 2025
This echoes natural concepts of data clustering viewed as information compression or lump- ing (Rosvall& & Bergstrom 2008 [011]] Lambiotte et al. 2014), and of clusters as equivalence classes (Brualdi) 2010). The MCbiF then aggregates all clusters (simplices) from partition ((s) to (()) through the union operators, such that a k-simplex = [21, [ -+++1 E K consists of ele- ments that are assigned to the same cluster (at least once) in the interval [s, t]. i.e., Fi,..., J E C for some cluster C E 0(r), r E [s. f]. The bifiltration depends not only on the lag |t s| but also on the starting scale s, and captures the topological autocorrelation in the sequence of partitions, see Fig. D We first show that the MCbiF is indeed well-defined bifiltration.
Proposition 4. .M is a multi-critical bifiltaaion uniquely defined by its values on the finite grid t) with partial order (s, t) S (8. ifs 2 & t S tt.
P = {(s,)) t) [f,,-..,ta. .tm. (tt))x((x....... tas]
The MCbiF leads to a triangular commutative diagram where arrows indicate inclusion maps between abstract simplicial complexes (Fig. D. The sequence of partitions ((t) is encoded by the complexes Kt on the diagonal of the diagram, hence the MCbiF is complete invariant of 8. Moving along horizontal arrows corresponds to fixing a starting scale and going forward in the sequence 6, thus capturing coarse-grainin along vertical arrows corresponds to fixing an end scale and going backwards in 0, capturing fine- graining. By fixing := 1 (top row in the MCbiF diagram), we recover the 1-parameter Multiscale Clustering Filtration (MCF) defined by Schindler & Barahona (2025). see Remark 24
The proof is straightforward. see Appendix B.2 2
Moving
Applying MPH to the bifiltration M at dimensions k < dim K, for K = Kftar, leads to tri- angular diagram of simplicial complexes Hy(K*:)) called persistence module (see Appendix E.2 We show in Proposition 27 in Appendix B.2 that the MCbiF persistence module is pointwise finie- finittl presentable and block block-decomposable (see|Botnn) Botnan & Lesnick (2023) for defini- tions). These strong algebraic properties are important because they guarantee algebraic stability of the MCbiF (Bjerkevik 2021). In particular, the finite presentation property implies stability of the MCbiF Hilbert functions HF (s. () (see Eq. 12 in Appendix F.2) with respect small changes in the module (Oudot & Scoccola [204 Corollary 8.2.). This justifies the use of HF's (s, t) as simple interpretable invariants for the topological autocorrelation captured by in the following.
3.1 MEASURING TOPOLOGICAL AUTOCORRELATION WITH MCBIF
We now show how the topological autocorrelation measured by the (F, 1) of MCbiF can be used to detect cluster-assigne assigmment conflicts. We focus on dimensions = 0, in this paper, where HFo(s, t) counts the number of connected components of Ks. and 1FF t) is the number of I-dimensional holes in K*tt (see Appendix E.2) We show below that the computation of these invariants reveals different aspects of the nin-hieryr in the sequence of partitions.
Low-order Non- Hierarerrryyyy in Sequences of Partitions An important aspect of hicrarchy is the nestedness of the clusters in the sequence. We say that 0 is nested in (s. t] when Vr,, r2 E [, have that V C' E ((r)), C" E ((r2), one of the sets C' C², C² CC or C E C' is empty. See Korte & Vygen 2012 Definition 2.12), We say that 0 is strictly nested when 0 is nested in [t1, oo). It follows directly that a hierarchical sequence is always nested. However, the converse is not necessarily true, as illustrated by the nested, non-hierarchical sequence in Fig. Zp. To quantify this low-order non-hierarchy in sequence 8, we can compute the invariant HF (FFo(s. t) with its associated notion of O-conflicts defined next. Recall that each partition 0(1) can be interpreted as an equivalence relation ~e where ~it y if C' E O(() such that 2r, y E C' (Brualdi) (2010).
Definition 5 (O-conffict and triangle O-conflict). a) We say that 6 has a O-conftict in [s, if the subposet ([ss, t)) has maximum, i.e., Dr [s, such that 0(r') < (rr). 'r' E [s. b) We say that 0 has a triangle O-conftict in [s, t] if 3 x, y, E X such that T2 E [s x ~r, ~r z and Ar [s, f); x oe y de
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Next, we show that all triangle O-conflicts are O-conflicts. Moreover, all O-conflicts break hierarchy, and triangle O-conflicts additionally break nestedness.
Proposition 6. (i) Every triangle O-conflict is O-conflict, but the opposite is not true. (ii) 0 has non-hierarchical in [s, tl. (iii) If I is either coarse- or fine-graining but
O-conffict in [s, f) then non-hiirarchical in [s. t). then 0 has O-conffict in (s, tl. (iv) If has triangle O-conflict in [s, t). then 0 is non-nested in [s, tl.
See Appendix B.2. for the simple proof. Fig. 2p illustrates O-conflict that is not a triangle O- conflict, and Fig. 2 shows a triangle O-conflict. The following proposition develops a sharp upper bound for HFo that can be used to capture O-conflicts.
(i)) (s) minre[s..] (8()) iff 0 has a O-conftict in [s, (ii) HF)(s,t) |0(r)) for r E [s. itf ((r)
Proposition 7. (i) HFu(s,t) t) S min.relat] (0(r)) Vis S oc).
the maximum of the subposer [[[s, t)).
See Appendix B.2.T for the proof. Proposition 7 shows that HF, measures low-order non-hierarchy in by capturing O-confficts. To quantify this, introduce global normalised measure for 0.
Definition (Average O-conflict). Let == LM + far-s The average O-conflict is defined as:
M-I
HF,(s,t) T - 41 A1 minrele. HF)(
s To (0)
ds dt S 1.
Higher values of co (0) indicate a high level of O-conflicts and increased low-order non-hierarchy, as shown by the next corollary.
Corollary 9. (i) If is hierarchical in [s, then (s) =0-mmni mmin((((s),, As a special case, this implies HF (t. = (0(t)). vt 2 ti- (ii) (0) > 0 iff 0 has a O-conflict. (iii) Let 9 be either coarse- or tim-igaining. Then, (0) = ift is strictly hierarchical.
See Appendix B.2.1 for the proof. Furthermore, can detect triangle O-conflicts by analysing the graph-theoretic properties of the MCbiF 1-skeleton k Proposition 30 in Appendix B.2. shows that clustering coefficient C(K).t) indicates the presence of a triangle O-conflict.
Higher-order Inconsistencies between Clusters in Sequences of Partitions Measuring 0- conflicts in / is only one way of capturing non -ierrhirrr An additional phenomenon that can arise in non-hierarchical sequences is higher-order inconsistencies of cluster assignments across scales. These are already captured by the 1-dimensional homology groups (Schindler & Barahona, 2025) and the associated notion of 1-conflict, which we define next. Recall the definition of 1-cycles Zi (K*++) and non-bounding cycles Hi (K*+), see Eq 14 in Appendix E.3 Definition 10 (1-conflict). We say that 0 has a I-conflict in [s. if 21,...,Ia E X such that the I-cycle = [m1, 322]
+ [In-11 + 21] E (Ko.)) is non-bounding; in other words, [x] E H((K*t) with [z] # O.
The number of distinct 1 1-conflicts in [s, t] (up equivalence of the homology classes) is given by HF t). We first show that 1-conflicts also lead to triangle O-conflicts and thus break hierarchy and nestedness of 8.
Proposition 11. (i) HF (s,t) 2 iff 0 has a 1-conflict in [s, t). (ii) If 0 has a I-conflict in [s, t), then it also has triangle O-conflict. (iii) If is hierarchical in (s, f). then HF 1F 0.
See Appendix B.2.1 for a proof. Proposition 1 shows that a 1-conflict a special kind of trian- gle O-conffict arising from higher-order cluster inconsistencies across scales. This is illustrated in Fig. I and Examples 37-38 in Appendix C which present sequences of partitions where different I-conflicts emerge across scales. Note that Proposition II (iii) shows that the MCbiF has a trivial 1-dimensional MPH if 0 is strictly hierarchical.
The presence of I-conflict in [s, signals the fact that assigning all the elements involved in the conftict to a shared cluster would increase the consistency of the sequence 9. Hence, when a 1-conftict gets resolved (e.g., the corresponding homology generator dies in the MPH at (s. t.), < t'), then (tt)) is called a conflict-resolling partition (Schindler & Barahona 2025). This is illustrated in Example 37 where a 1 I-conftict gets resolved by ((4) = 1, and Example 38
Remark 12.
where three different I-conflicts get resolved one by one by partitions (77,, 0(8) and (()).
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To quantify the presence of 1-conflicts in 0, we introduce unnormalised global measure.
Definition 13 (Average 1-conflict), Ler T be as in Definition
The average 1-conftict is defined as:
2
T
[-
(0) - T 112
HF HF, dt.
(3)
Corollary 14. (0) > 0 has a
a I-conflict. In particular, if is strictly nested, then ei (0) = O.
Figure in Appendix A summarises all of our theoretical results and their relationships.
3.2 MCBIF AS HIGHER-ORDEA SANKEY DIAGRAM
Consider the definition of the Sankey diagram of the sequence of partitions provided in Section E. and its associated representation as an . lfylaye graph with vertices Vn each layer representing the clusters of ((4m). see Eq 9 Let define the disjoint union A(e, m) == Viw.. VV, 1 S l S which assigns an index to each cluster 0(1) for E [4,, bm]. Furthermore, recall that O(); denotes the i-th cluster C₁ of (tt). The nerve-based MCbiF can then be defined as follows.
Definition 15 (Nerve-basee MCbiF), Lets E [te, te+1), f= 1..,...MM .....,,M.. M-1, E [tm, (antmm+)),
M or 2 Im for m M. We define the nerve-bused MCbiF as
M =
where Kst := for S A(e,m)
Ω 0(L)) (n,
(4)
The nerve-based MCbiF M is a 1-critical bifiltration with simplices representing clusters and their intersections, in contrast the original MCbiF M (Eq. [ in which simplices represent elements in X and their equivalence relations. Despite these different perspectives. Proposition 34 in Ap- pendix B 2.2.2 shows that M and M lead to the same MPH and can be considered as equivalent. However, the dimensionality of M and M can differ, as shown in the following proposition.
(a) No conflict
Proposition 16. (i) dim Kttt max maX |C| Vt, < S t. (ii) dim K Ktmdmmn n, V1 S m M, n A M
(
(i #2. 2a.
See proof in Appendix B.2.2 The nerve-based MCbiF is therefore computationally advantageous when the number of
(b) O-confict
scales M is smaller than the size of the largest cluster, M <
22
{2a)
(*6.
(axc) oo)) IC). The serve-based M can be interpreted as a higher-order extension of the Sankey diagram S(0) (Eq. 9) Yet, unlike S(O), which only records pairwise intersections between clusters in consecutive partitions of 6, M also ac- counts for higher-order intersections between clusters in sub- sequences of f
(24)
(21)
{
(zu)
(c) Trianggaa D-contiict
{x))
(m
{#s za)
(.)
Proposition 17. The Sankey diagram graph S(0) is a strict I-dimensional subcomplex of K := KtidM In particular, Vm tmmm and Em Kmmmm1 Vm 1...., M Hence, we can retrieve (0) from the zigzag filtration
(x.)
(4) 1-conflict
1.
(x),2.)
Rhh.m - klmimm- e Rlm4smm+1
(5)
(#2.
R4
V
Vn+1
which is subfiltration of the nerve-based MCbiF.
Figure 2: Characterisation of con-
flicts in a single-layer Sankey dia- gram, following Proposition 18
See proof in Appendix B.2.2 For details on zigzag persis- tence, see Cerlsssonn & de Silva 2010) and Appendix E.4 Next,
we characterise the ()- and I-conflicts that can arise in a single layer Em, of the Sankey diagram.
Proposition 18. (i) There is O-conflict [tm, tm Eu E Vm, and u E Vm+1 with deg(u) 2 and deg(t) 2. where deg denotes the node degree in the bipartite graph (Vn, W Vnt1i Em associated with the Sankey diagram S(0). (ii) There a triangle O-conflict in [tms tmtal iff there is path of length at least in Fmn (iii) There is I-conflict in [I (m-1) 1) iff there cycle in Fm..

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See Appendix B.2.2 for a proof and Fig. 2 for an illustration. Importantly. eycle in Em leads to crossing in Em that cannot be undone Fig. 2t. Hence, Proposition 18 (iil) implies that the sum of
the elements of the superdiagonal of HF provides lower bound for the minimal crossing number of the Sankey diagram, Ko defined in Eq. 10
Corollary 19. Em=1 HF (tmm fm++1 Re.
Remark 20. Note that 1 I-confticts that arise across multiple partitions in the sequence (i.e., across multiple layers of the Sankey diagram) do not necessarily lead to crossings. See Fig. foor a Sankey diagram with no crossing despite the presence of I-conflict. However, we hypothesise that the full HF' and HF y feature maps capture more complicated crossings that arise in the Sankey layout across many layers. This insight is exploited in our compulational tasks below.
4 MATHEMATICAL LINKS OF MCBIF TO OTHER METHODS
Ultrametries. Given sequence 0 with 0(1 = 0) Ô and (((M) = 1, let us define the matrix of first-merge times conditioned on the starting scale
(o, my)) min 2 3 0(t) I E C}.
(6)
When 8 = O, this recovers the standard matrix of first-merge times Dg == Do.o in Section 2 If hierarchical, i.e., agglomerative dendrogram, then D3 is ultrametric. as it fulfils the strong triangle inequality: D2 2) I max (z, y). Dø(z, 2)) Vzz y, E From Corollary 9 we have that the number of branches in the agglomerative dendrogram at level t. which is given by ((t) is equal to HF (s. 1) for any < f. Hence, HF 1) contains the same information as the ultrametric in the hierarchical case, see [Schinllrr Barahona (2025) and Proposition 35| If, on the other hand, 0 is non- non-hierarchica,, triangle O-conflicts can lead to violations of the (strong) triangle inequality.
Proposition 21. The triplet x, y, 2 E X leads to a triangle O-conflict in [s, f] isf x. y; violate the strong triangle inequality for D8,s. i.e., Dø, (2, 2) max(D (z. 3). D9,5 (y) z)).
See Appendix B.3 for a proof and Fig. 7h for an illustration. Proposition 21 shows that (0) measures how much the ultrametric property of Dg is violated. Recall that Dg.e is a dissimilarity measure that can be used to define a filtration (Chazal et al. 2014). We show in Proposition 35 in B.. Appendix B.3 that the O-dimensional MPH of MCbiF corresponds to the O-dimensional MPH of the Merge-RRps bifiltration constructed from Dø..- In the hierarchical case, the Merge-Ripp Pirge-Rips bifiltration has trivial 1 I-dimensional MPH, as D3 fulfils the strong triangle inequality, and is thus equivalent to the MCbiF, whose I-dimensional MPH is also trivial in the hierarchical case, Proposition 11
Conditional Entropy. The conditional entropy (CE) is only defined for pairs of partitions as the expected information of the conditional probability of (tt given ((s), denoted Pals (see Eq. 15 in Appendix F.. For the case M 2 (i.e., only two partitions in the sequence 0), HF (2) follows directly from the spectral properties of the matrix Ptg t1 tzlt,
PTT interpreted as an undirected graph.
Proposition 22, HF t2) = dim(ker L) for graph Laplacian L = diag( (oag(Peatt,)) Ptgle. - PT falts
See Appendix B.3 for proof. Note that Pri. only encodes the pairwise relationship between clus- ters, and does not capture higher-order cluster inconsistencies. particular, CE cannot detect 1- conflicts arising across more than two scales, seen in Example 39 in Appendix
5 EXPERIMENTS
5.1 REGRESSION TASK: MINIMAL CROSSING NUMBER OF SANKEY LAYOUT
Our first experiment considers a task of relevance in computer graphics and data visualisation: min- imising the crossing number of Sankey diagram layouts (Zarate al.) 2018 Li et al.) 2025). This minimisation is NP-complete. Here we use our MCbiF feature maps to predict the minimal crossing number F (Eq. II of the Sankey diagram S(A) of a given sequence of partitions (see Section E.
We build synthetic datasets by sampling randomly from II the space of souase-graining sequences of partitions of N elements with M change points, see Definition 40 in Appendix [.1] Setting
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M = 20, generate two datasets of 20,000 random sequences I for N and N = 10. For each we compute five feature maps: CE, ARI, MOD (see Section 2), and our MCbiF Hilbert functions (HF, and HF,r 1). To benchmark against representation learning, also consider the (non- unique) raw label encoding of given by the N x M matrix whose mn-th column contains the labels of the clusters in m() assigned to the elements in X, and the Sankey graphs S(0) (Eq.. 9 Our prediction target is = Ke (Eq. [[], the minimal crossing number of the layout of the Sankey diagram, as computed with the OmicsSankey algorithm (Li et al. 2025). See Section E. for details. We expect that predicting y becomes harder for N = 10 because the increased complexity of I allows for more complicated crossings in the Sankey diagram.
As preliminary assessment, we compute the Pearson correlation, between the crossing number y and summary statistics of the five feature maps under investigation: the consensus indices VI, ART and MOD, and the MCbiF average measures Gg and ē₁ The correlation between the consensus indices and y is low, higher for Co. and highest for é (r 0.47 for N 5. 10) (see Fig. [T]
in Appendix D.I). This is consistent with our theoretical results in Section 3.2 which show the relation between the crossing number and HF (see Corollary T9p.
Table 1: Regression task. Test R² score of LR, CNN. MLP and GCN models trained on different feature sets for N and N 10. See Appendix D.I for confidence intervals and train R² scores.
Raw label encoding
Sankey graph
Method
HF.
HF1
HFo & HF;
CF
ART
MOD
LR 0.078
0.147
0.486
0.539 0.544
0.392
0.166
0.413
5
CNN
0.267
0.155
0.504
0.492
0.422
0.354
MLP
0.104
0.150
0.491
0.541
0.409
0.214
0.351
GCN
0.416
LR 0.038
0.214
0.448
0.516
0.457
0.246
0.345
10
CNN
0.072
0.211
0.448
0.507
0.454
0.294
0.312
MLP
0.036
0.212
0.450
0.514
0.458
0.256
0.246
GCN
0.229
For the regression task of predicting Rg. we split each dataset into training (64%), validation (16%) and test (20%). For each feature map (or their combinations), we train three models: linear re- gression (LR), multilayer perceptron (MLP), and convolutional neural network (CNN). We use the mean-squared error (MSE as our loss function and employ the validation set for hyperparameter tuning. See Appendix D.I for details. We then evaluate model performance on the unseen test data using the coefficient of determination (R²). As representation learning from the raw data, we also train models (LR, CNN, MLP) on the raw label encodings, and a graph convolutional network (GCN) on the Sankey graph S(0) (Eq. D. [ that models trained on MCbiF feature maps outperform all representation learning models; even the GCN trained on Sankey graphs (J2² 0.416 for N 5; R² 0.229 for N = 10) has significantly lower performance (p < 0.0001, Wilcoxon test residuals) than simple LR model based on HF, (R² 0.486 for N 5; R² 0.185 for N = 10). This reflects the fact that the MCbiF is a higher-order extension of S(8) that better captures the global properties of (see Proposition 17).
The MCbiF feature maps also outperform the baseline feature maps CE, ART and MOD; the com- bined HF, and HF, features (R² = 0.544 for N = 5; R 0.516 for N = 10) have significantly better performance (p < 0.0001, Wilcoxon test) than the best baseline features CE (12² 0.492 for N 5; n22 0.458 for N 10). Variability analysis (bootstrapped 95% confidence intervals) shows that models trained on the combined HF, and HF, features outperform the other models (see Table 3 in Appendix D.J). Note that the strong performance of simple LR models underscores the interpretability of the MCbiF features, important for explainable AI (Adadi & Berrada, 2018).
5.2 CLASSIFICATION TASK: PREEEERVIP SEQUENCES OF PARTITIONS
In our second task, we classify whether a sequence of partitions is order pnder-preserving. i.c., whether is compatible with total ordering the set X, Definition 41 in Appendix D.2 This task is relevant in socio-cconomics (preference relations in utility theory in social sciences (Roberts) (009) and computer science (weak ordering and partition refinement algorithms (Habib al. 1999).
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We carry out this classification task synthetic data for which we have ground truth. From the space of coarse- coarse-graining graining sequences partitions I introduced in Definition 40 we generate a balanced dataset of 3,700 partitions
I half of which order-preserving (y () and the other half are non-order- non-orddr--peeeerving preserving (y- = 1). The loss of order-preservation is induced by introducing random swaps in the cluster assignments within layers. See Appendix (D.2 details. For each of the generated 0 we compute CE, ARI, MOD, HF' and HF', using the computationally advantageous nerve-based MCbiF. We choose N 500 and M
30 demonstrate the scalability of our method.
Whereas we find no significant difference between the baseline consensus indices of order-preserving (y = 0) and non-order- 1) sequences, we ob- serve a significant increase of 6g and e for order-preserving sequences (Fig. 12 in Ap- pendix D.2). For the classification task, we split our data into training (80%) and test (20%). For each feature map, we train logistic regression the training split and evaluate the accuracy on the test split, see Appendix [.2]. We find that HF, predicts the label = {0, 1} en- coding the (lack of) order--reservation with high accuracy (0.97). In contrast, CE, ART, MOD and the raw label encoding of do not improve on a random classifier (Table 2) Our performance variability analysis (bootstrapped 95% confidence intervals) confirms that the model trained on HFj consistently outperforms all models (Table in Appendix This demonstrates the high sensitiv- ity of MCbiF order-preserver in 0, due to pon-orrer-preserving sequences inducing 1-conflicts that are captured by HF,
Table 2: Classification task. Test accuracy of logis- tic regression trained on different features. See Ap- pendix D.2 for confidence intervals.
preserving (y
Raw label encoding
HFO
HF,
CE
ARI
MOD
0.53
0.56
0.97
0.50
0.49
0.46
H5/5
HESA
(b),
50
50
20
7
0.6
12"0
127
107 2.7
Ty Tg
Figure 3: (a) Analysing non-hierarchical sequences of temporal partitions O,, compiled from social interactions of a mice population over 9 weeks. Each 0z, is sequence of social groupings 0z, (t) observed over week Different sequences of partitions are computed
function of the parameter Tú. Sankey diagrams and MCbiF feature maps are shown for 0-, at T; (i = 2, 4. 8), identified as robust Bovet al. 2022). These three sequences exhibit different types of non-hierarchy, as shown by our topological feature maps and measures of average O-conflict (og) and average 1 1-conflict (τ1)- (b) The robust 0,, (i = 2, 2,4, 8) found in Bovet et al. (022) correspond to distinct topological characteristics of the sequences of partitions, between MCbiF Hilbert functions. The Hilbert distances also capture increased time reversibility in 0,, and fr,, due to the larger stability of social groupings over time (see Fig. 14 in Appendix D.3
captured by the block structure in the distance
5.3 APPLICATION TO WORLD TEMPORAL DATA
In our final experiment, we apply MCbiF to temporal sequences of partitions computed from real- world contact data of free-ranging house mice. This data captures the changes in the social network structure of the rodents over time (Bovet et al. 2022). Each partition 0, (t) describes mice social groupings for N = 281 individual mice at week t E [1...., 9) for the nine weeks in the period 28 F May 2017, so that the sequence 8, captures the fine-grain mnn-graining of social groups from win- ter to spring. A partition sequence is computed at temporal resolution T > O, where the parameter t modulates how fine the temporal community structure is (Fig. 13). See Bovet al. 2022) and Ap-

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pendix D.3 for details. We then compare temporal partition sequences 0,, for different resolutions To, 2 = I........ 9, as given in Bovet et al. (2022). For each 8,,, we obtain HFg and 1F1 using the nerve-based MCbiF, which induces 50-fold reduction in computation time due to much lower number of simplices (260 simplices for the nerve-based MCbiF vs. 1 116,700 for the original MCBiF).
Bovet al. (2022) identified that the temporal resolutions 72 = 1s, 7a 60 and 75 24 h lead to robust sequences of partitions. Using the Hilbert distance (i.e., the Ln-norm on the 0- and I-dimensional MCbiF Hilbert functions), we find these temporal resolutions to be representative of three distinct temporal regimes characterised by different degrees of non-hierarchy, as measured by Co and 61 (Fig. B). In particular, high Eg indicates that mice tend to split off groups over time, and high ε indicates that mice meet in overlapping subgroups but never jointly in one nest box. Note that 0-, has a strong non-hierarchical structure because the large-scale mice social clusters get disrupted in the transition to spring. In contrast, 07. is more hierarchical as it captures the underlying stable social groups revealed by the higher temporal resolution. Yet, P,, has the strongest hierarchy as indicated by lower Eo and an absence of I-confficts (č1 = 0) and thus corresponds to a sweet spot in hierarchical organisation between low and high temporal resolutions. These findings also underpin the optimised Sankey diagrams in Figure Bh. While the Sankey diagram for 0z, can be drawn without any crossings, the optimal Sankey diagram for fr, has a crossing from week to week 2, which resembles the 'hourglass' 1-conflict (Fig. (1)). This is predicted by our MCbiF features: HF, (1, 2) = for 0r. implying one crossing between the first and second layer of the Sankey diagram that cannot be undone, as per Corollary 19
CONCLUSION
We have introduced the MCbiF, novel bifiltration that encodes the cluster intersection patterns in multiscale, non-hierarchical sequences of partitions. Its stable Hilbert functions HF', & quantify the topological autocorrelation of the partition sequence @ and measure non-hierarchy in two comple- mentary ways: at dimension & = it captures the absence of maximum with respect to the re- finement order (0-confficts), whereas at dimension k: = 1 it captures the emergence of higher-order cluster inconsistencies (1-conflicts). This is summarised by the measures of average O-conflict čp (0) and average 1-conffict ε₁ (8), which are global, history-dependent and sensitive to the ordering of the partitions in 0. The MCbiF extends the 1-parameter MCF defined by Schindler & Barahona (2025) to 2-parameter filtration, leading to richer algehraic invariants that deseribe the full topological information in 0. It is important to remark that the MCbiF is independent of the chosen clustering algorithm and can be applied to any (non-hierarchical) sequence of partitions 0. We demonstrate with numerical experiments that the MCbiF Hilbert functions provide topological feature maps for downstream machine learning tasks, which are shown to outperform both baseline features and representation learning on the raw data for regression and classification tasks on -irraieralllaal sequences of partitions. Moreover, the grounding of MCbiF features in algebraic topology enhances interpretability, a crucial attribute for explainable Al and applications to real-world data.
Limitations and future work Our analysis of the MCbiF MPH is restricted to dimensions 0 and I for simplicity. However, the analysis of topological autocorrelation for higher dimensions would allow us to capture more complex higher-order cluster inconsistencies and could be the object of future research. Furthermore, we focused here on Hilbert functions our topological invariants because of their computational efficiency and analytical simplicity. which facilitates our theoretical analysis. In future work, we plan use richer feature maps by exploiting the block decomposition of the MCbiF persistence module, which leads to barcodes (Bjerkevik 2021). or by using multipa- rameter persistence landscapes (Vipond 2020). Another future direction to use MCbiF to evaluate the consistency of assignments in consensus clustering (Strchl & Ghosh| 2002 Vega-Pons & Ruiz-
Shulcloper 2201 1). It can be shown that the values of the Hilbert function HF (8, t) further away from the diagonal (s = f) are more robust to permuting the order of partitions in Ø (see Proposi- tion in Appendix B.4]. In particular, HF (t tM) only depends on the set of distinct partitions in (((ti, oo)) and is independent of any permutation in their order. Hence, in future work, HF (t,, (m) could be used as an overall measure of consistency in 0 in the vein of consensus clustering. Finally, minimal cycle representatives of the MPH (Li et al. 202 can be used to localise I 1-conflicts in 8, which is of interest to compute conflict-resllvigg partitions in consensus clustering, or to identify inconsistent assignments in temporal clustering (Liechti & Bonhoeffer 2020).
&F&(s,
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REPRODUCIBILITY STATEMENT
Detailed proof' of all theoretical results can be found in Appendix and extensive documentation of our experiments in Appendix An implementation our method and code to reproduce our numer- ical experiments is available at: https://gtt github.uhuuu.. com barm/barahonaaaa research group/ MCbiF, The dataset studied Section 5.3 is publicly available at: htttps://datavrrs
harvard. edu filee. xhtm 2
ACKNOWLEDGMENTS
IS acknowledges support from the EPSRC (PhD studentship through the Department of Mathe- matics at Imperial College London). MB acknowledges support from EPSRC grant EP/N014529) supporting the EPSRC Centre for Mathematics of Precision Healthcare. This work benefited from discussions at the Asem Alaa, Christian Schindler and Jacopo Graldi for valuable discussions computational as- pects this project, Alex Bovet for discussions on the temporal dataset studied in this paper, and Arne Wolf for discussions the stability of multiparameter persistence modules. We also thank Heather Harrington for helpful discussions and for the opportunity to present work in progress at the Max Planck Institute of Molecular Cell Biology and Genetics Dresden (MPI-CBG) in July 2025.
TDA seminar November 2024. We thank Kevin Michalewiez,
AUTHOR CONTRIBUTIONS
IS developed the mathematical theory, conducted all numerical experiments, and authored the initial draft of the paper. MB provided supervision, idea generation and guidance throughout the full research process. Both authors wrote and edited the final manuscript.
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APPENDICES
A
SUMMARY OF KEY THEORETICAL RESULTS
In Figure 4
we provide summary of our theoretical results and their relationships.
Higher-order
Proposition 11
Remark 28
Inconsistencies in [s,t]
Non-nested in |s,
Non-hierarchiccal in [s,
Proposition 6,
Progesition Remark 29
Datinition
Remark 29
Proposition 11
Proposition
1-conflict in [s.t]
x
Triangle 0 O-conflict in [s,
x
O-confiict [s,
Definition
Proposition
Proposition
Proposition
1-dim. Hilbert function HF (s, 2
Strong triangle inequality of Dy,, violated
Clustering Coefficient
O-dim. Hilbert function HF HFo(s, () < min [8())
C(K]")<< <
Corollary 14
Corollary
Average 1-conffict (0) > 0
Average 0-conflict co( > 0
Figure 4: Summary of key theoretical results and their relationships. Double-headed arrows repre- sent equivalences (iff), whereas hingle-heade arrows represent implications (if).
B PROOFS OF THEORETICAL RESULTS AND ADDITIONAL DETAILS
B. PROOFS AND DETAILS FOR SECTION
We first state a simple fact about coanse-graining sequences of partitions.
Remark 23. Let (t);; denote the i-th cluster C; of ). It is a simple fact that 0 is coarse-grainee poause-grainingg iff
the mean cluster size non-decreasing. i.e.. [ Σ [0()] [0(s); S [o(E]] Σ, (0(+)] [0(t), for S
The proof follows directly from the fact that Σ [0()) 10(s).l = Σ [t)); = N.
B.2 PROOFS AND DETAILS FOR SECTION
We provide proof for the multi-criticality of the MCbiF filtration stated in Proposition
Proof of Proposition 4 The MCbiF is indeed bifiltration because K K''.'' if s' and t t t'. See 5 for the triangular diagram of the MCbiF filtration, where arrows indicate inclusion maps. The MCbiF is uniquely defined by its values on the finite grid [t,....,tal
[ti...., tvl because 0 has change points t < ... < fM- It is multi-critical bifiltration because for x E X we have [x] E Kan for all 8, E [t,, oo) [41, oo). In particular, x E T and x E K for 5 > 0 but (t1, (1) and (t1 + 5, 1 + 8) are incomparable in the poset [f1, oo) x [t1,
L
Remark 24. By fixing s == t1 (i.e., the top row in the commutative MCbiF diagram), we recover the 1-parameter Multiscale Clustering Filtration (MCF) defined by Schindler & Barahona 2025). MCF was designed to quantify non-hierarchies in coarse- gransiigrainingggg sequences of partitions and thus cannot capture fine-graining For example, large cluster C E 0(s') prevents MCF from detecting cluster assignment conflicts between elements x, y E C l 2 8', see Section In particular, the MCF is not a complete invariant of O. In contrast, MCbiF is complete invariant of 0 and encodes the full topological autocorrelation contained in 0 by varying both the starting scale s and the lag t
Next, we provide formal definitions for algebraic properties persistence modules, see Botnan & Lesnick 2023) for details.
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J
K Kit
Kie
I
Ktd
- Kt''"
Kt"
Figure 5: Triangular commutative diagram of the MCbiF for 1 S S S th. The arrows indicate inclusion maps between simplicial complexes.
Definition 25. For partially ordered sets Pı, P2, we call an interval I Pi X P2 a block if it can be wrillen one of the following types:
1. Birth quadrant:
S1 x for downsets S C Pi and S2 C P2.
2. Death quadrant: = S1 x for upsets S1 C P1 and S2 C Ps.
3. Vertical band: == S1 x P2 for an interval S C P1.
4. Horizonial band: I = Pi x Sa for an interval S2 S P2.
Definition 26. Let Vect denote the category of k-vector spaces for a fixed field k. For partially ordered set P. P-indexed persistence module is a functor F Vect. We say that:
a) F is called pointwise finite minte-dimensional if dim(Fa) < do for all P.
b) F is called finitely presented if there exists morphism of free modules Φ1 Fi F1 such that coker( (1)) = and Fo and F1 are finitely generated.
c) F is called block-decomposable if it decomposes into blocks F D BEB(F)) KB where B(F) is multiset of blocks (see Definition 25 that depends on F.
which shows that the MCbil persistence mod- ule (see Fig. 6 is pointwise fiite-dimensionall finitely presented and block-decomposable The proof relies on the equivalent nerve- nerve-based construction of the MCbiF (see Proposition (34], and the exactness of the persistence module from which block- boockdccomposabillyy follows (Cochoy & Oudot 2020).
We can now provide the proof for Proposition 27
Proposition 27, For any S dim K, the MChiF persistence module Hg(KK.)) is pointwise finite- dimensional, finitely presented and block-decomposable
The MCbiF module pointwise umite-dimensiona because the homology groups of finite simplicial complexes are finite. As the MCbiF is defined uniquely by its values finite grid (Proposition 4 its persistence module consists of finitely many vector spaces and finitely many linear maps between them, hence it finitely presented.
Proof of Proposition 27
To prove dleck-decomposabiiiiyy same persistence module, see Proposition 34 As the module is uniquely defined by its values on finite grid, if the persistence module is exact. Hence, it suffices to show that for all t diagram
the nerve-based MCbiF (K*,)),, Sast which leads to the
Theorem 9.6 by Cochoy & Oudot (2020) that implies dlok-decooposabbiliy
< t s t' s t" s t" the
(a(k(()) - He(K())") T (K"""")) Ha (Kt
induces exact sequence:
H (e( Ha(k?l)" Ha(l""("") (K(l"
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By construction of the MCbiF, Kttt K U Furthermore, k Kit' U UK''"" and Kt =t "Kt" ,t" i is piecewise-constam function with change points i <... t we can assume without loss of generality, t = 4. = le, t" = l.m t" = in for change points L < < 1m < in of 0 such that A(k, E) n A(m, n) = 0. Hence, Ki,e" n Ret," = 0 and
Kt = K KKt"". This means that Eq. equation is Mayer-Vietoris Victoris sequence for all
k 2 O, implying exactness (Hatcher 2002 p. 149) and proving the block decomposability (Cochoy
& Oudot 2020 Theorem 9.6).
HK(K Ha(K T
H((K("))
H((K'N)
Ha(Ked"))
Ha(K""())
Figure 6: Multiparameter persistence module of the MCbiF for t S S t' S t". The arrows indicate linear maps between vector spaces.
B.2.1 PROOFS AND DETAILS FOR SECTION 3.1
Remark 28. follows directly from the definitions that hierarchical sequence 0 is always nested. However, nestedness does not necessarily imply hierarchy, as illustrated by the example in Fig. 2
We continue with the proof of Proposition 6 that relates O-conflicts to hierarchy and triangle 0- conflicts to nestedness.
Proof of Proposition 6 (ii) If has a O-conflict then 3r1, r2 E [s, such that ((r1) I ((r2) and ((r1) ((r2), otherwise 0[[s, f)) would have maximum. Hence, 6 is not hierarchical in [s.
(iii) Let us first assume that is coarse-graining, i.c., [0(t) S (0(r) for all E [s, f.. We show that O-conflict in [s, implies that 0 is hierarchical in [s, f) Let 71,72 E [s. with rı r2, then the subposet (((r1, r2]) has a maximum because of the absence of a O-conflict, and the maximum is given by ((r2) due coarse-graining, Hence, 0(r1) 0r22). As T2 were chosen arbitrarily, this implies that 0 is hierarchical in [s, t). The argument is analogous for the case that 0 is fine-graining.
(iv) Let y, E X be in a triangle O-conflict. In particular, mrn y/ ~rg z with xr y, x f z and y * z. Hence, there are C' E 0(r1) and C' E 0(r2) such that &,, E C C'and y, = E C", as well as 2 & C and x & C'. This implies {a) E C C², {2} C' C and {v} E CnC' showing that C and C" are nenenes.ed. Hence, 0 is non nos-nested in ts.
i) Moreover, Ar E [s, f] such that ~, y. particular, Dr [s, such that D0" E ((r) with C C" and C C C"i Hence, 3r E [s,t] t) such that ((r1) < ((r) and 0(r2) < 0(r), implying that the subposet {[s, f has no maximum. This shows that every triangle O-conflict is also a O-conflict,
proving statement (i). Note that the opposite is not true, as illustrated by the example in Fig. 2p. O
Remark 29. Non-nestedness and non-hierarchy do not imply the presence of a O-conflict. To see this, consider the simple counter-exampll given by 8(0) {(z.v) v). {2}}, A(1) = i. 0(2) {[)), (y. 2)}, which is non-nested but the partition ((1) the maximum of the subposet ( 1, 2)). This illustrates the need for the additional assumption of coarse- or fine trainigg of 0 in Proposi- tion ( for the condition of no O-conflict to imply hierarchy.
We now provide a proof for Proposition 7 on properties of the O-dimensional Hilbert function of the MCbiF.
Proof of Proposition Z
(i) HF t) is equal to the number of connected components of Ks.t Let 10(r")] minyelaa) [0(r). We can represent 0(r) (Ci,..., C.) and
r' [s, () such that
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by construction AC E K for all C' A(r). Hence, if two elements z E X are in the same cluster C' E 8(r) then [z, y] E Kd* and the O-simplices [x], [y] E K are in the same connected component. As 0(r) has mutually disjoint clusters, this means that there cannot be more than disconnected components in K and (F 1) & c= - =0(r')) As As E [s, i) was chosen arbitrarily, this implies HF & min.e [0(r).
We prove statement (ii) by the contrapositive and show that the following two conditions are equiv- alent:
C1: HF = min,e |0(r)]-
C2: Br [s. t| such that 0(r') S 0(r), Vr' [s.
Note that C2 is equivalent to there is no O-conflict in [s,t]."="
"F" consider first that C2 is true and
((r) is upper bound for the partitions ((r'), r' ts. This implies that Vr' [s. f we have that VC" E ((r') there EC E ((r) such that C" E C. By construction of the MCbiF (Eq. T this implies Wa' E k there 3o E K such that o' a. This means K C K and thus K K has [0()] disconnected components this implies HF 1) = (0(r),, showing C1.
K As
=="To "To prove the other direction, assume that C1 is true. Then there exists C = t)) = [0(r) with [0(r)] = minge(s.s) [0(z)-- In particular, the disconnected components of Ka, are given by the clusters of ((r) denoted by Cı, .....,C.. C.. Let E [s, t] and C" E ((r). Then
[s, 1] such that
Di [I, such that C" _ C, 0(r) because otherwise the solid simplex AC' would connect two solid simplices in (AC1..... ACe]. contradicting that they are disconnected in Ks.f. Hence, the clusters of ((r') are all subsets of cluster of ((r). implying A(r') I ((r). As r' E [s, tl was
chosen arbitrary this shows C2.
We finally prove statement (iii). "=" Note that Hs) min," [6(r') according to (i). Then (ii) shows that C2 is true for r, i.e., H(r) is the maximum
|0(r)] implies |0(r)
of the subposet ([ss)t). "#" The other direction follows directly from the proof of (ii).
We next prove Corollary 9 about some properties of the average O-conflict, which follows immedi-
ately from Proposition 7
Proof of Corollary y We begin with the proof of statement (i). If hierarchical in [s, 8] then θ is either coarse- or fine-graining. Assume first that 0 is coarse- graining, then 0(s') < (()) for all s' e [s.t] and together with hierarchy, this implies that ((t) is an upper bound of the sub- poset 0[1s, d).. Hence, Proposition
7 (iii) shows that HF () |0()))- Moreover, HFo(s,!)
min( (0s)) ) because noarse-graining implies [0(=)] 2 (0(1))- similar argument also shows HFo ((s) mi)( ((t))) if 0 is fine-grainee graining.
We continue with proving (ii). (0) > 0 is equivalent to Bs,t E tu] such that HFo t) < minyefs.4] (O()), according to Definition g This is again equivalent to fs,t E [f1,M]] such that 0 has a 0-conflict in [s. (]. according to Proposition (i)).
We finally prove statement (iii). "=" (0) means that has no O-conflict in [4, oo). As is also
coarse- or finge-graining, Proposition 6 (iii) then shows that 0 is strictly hierarchical. "<"" If 0 strictly hierarchical, then has no O-confficcts according to Proposition (ii) and statement (ii)
implies that c0) 0.
We can further detect triangle O-conflicts by analysing the graph-theoretic properties of the MCbiF I-skeleton K Recall that the clustering coefficient C of graph is defined as the ratio of the number of triangles to the number of paths of length 2 in the graph (Luce & Perry 1949 2018).
Newman
Proposition 30. C(K)')) < 1 iff there is triple x, U E X that leads to a triangle O-conflict for [s. t). and which is not a cycle, i.e., additionally to property b) in Definition S we also have Arg E [8, {): x rgg z.
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Assume that C(K 1. Then there exist z, 2, X that form path of length but no triangle, see Newman 2018) for details on the clustering coefficient. Without loss of generality, [a, y], [y, x] E Ki but [z, 2 & Ktt This implics 3r1, r2 E [s, t]: x mrs rr and Dr E [s, t]: x ~r Hence, I. y/ 2 lead to triangle O-conflict.
Proof of Proposition 30
Let us consider the graph generated as the disjoint union of all clusters from partitions in [s,t] as cliques. This graph is equivalent to the MCbiF I-skeleton Ki Proposition 30 shows that the clustering coefficient of this graph can be used to detect triangle O-conflicts that are not cycles. To be able to detect triangle O-conflicts that correspond to non-bounding cycles, we turn to the 1-dimensional homology.
We next prove Proposition on 1-conflicts.
Statement (i) follows directly from the definition of I-conflicts that HF, (s, t) = dim H [<)])] 1 i @ has a 1-conftict.
Proof of Proposition 77
We next prove statement (ii: If (s, t) 2 there exists a 1-cycle = = [x1. x21 +..++++. [ann-1 Fn ]++ [x,, ø1] that is non-bounding. i.e., h := [a] * 0 in H1 (K**), see Appendix E.3 for details. Case I: Assume fr E [s. σ1 ~y 22 ~r 23. then it follows immediately that #1 22 2g lead to a
triangle O-conffict. Case 2: Assume Ar E [s, f): 21 ~ip 32 mep 23. As [=]
there exists 1-cycle = [81,, [2] + ................................ + +[-m-1, Fm] + [ām, Ø1 E (K(K)))) such that is homologous to 2, i.e., = + 0zw for w E (2(K))) and such that Dr [s, 8]: Tt map 32 mep Tz. In particular, 3,, 22 33
lead to triangle 0-conflict.
We finally prove statement (iii): If 0 is hierarchical, then it has no O-conflicts according to Corol- lary 9 Hence, 9 also has no triangle O-conflict in [s, t] and so (i) implies that HF 1s, t) 0.
O
Remark 31. Proposition 71 states that every 1 1-conflict is a triangle O-conflict. However, not every (triangle) O-conflict is I-oofflict, Example 37 Note also that several triangle O-conflicts in the sequence @ can lead to a I-conflict, when the triangle O-conflicts are linked together in such a way as to form a non-bounding cycle, see Example 37 We can lest for these systematically using HF
Remark 32. While O-conflicts (Cu(0) > 0) can be defined in relation to the refinement order that gives rise to the partition lattice, the partition lattice cannot be used detect eigher-order cluster inconsistencies (I-conflicts), which can be captured and quantified instead by HF, and the average measure 61 (0).
B.2.2 PROOFS AND DETAILS FOR SECTION
To prove the equivalence between the MPH of the MCbiF and nerve-based MPH, we can use Persistent Nerve Lemma for abstract simplicial complexes by Schindler & Barahonal (0225).
Lemma 33 (Schindler& & Barahona 22255) Lemma 32). Let K C K' be two finite abstract simpli- JKaJaEA cial complexes and {Ka JaEA A and [Ka be subcomplexes that cover K and K' respectively, based the same finite parameter set such that Ka C K2 for all A. Let K denote the nerve N ([]KalBaeAA a) and K' the nerve N (q)K) (aee)) the intersections Di= |Ka,l and n K are either empty contractible for all k E N and for all oo...., Or A. then there exist homotopy equivalences K- + |K] and K' + |K* that commule with the canonical inclusions |K| + [K"] and K- <'
See Schindler & Barahona
2025
Lemma 32) for a proof.
Next, we provide the proof about the equivalence between MCbiF and nerve-based MCbiF. Proposition 34. The bifiltrations M and M lead to the same persistence module.
Proof. We use Lemma 33 to show the equivalence between MCbiF and nerve-based MCbiF. For > s' and f < I', we denote K = k and K Ka.t, and further denote K' := K' and K" == 5.. As is a seccewise-constane function with change points t1 < ... < ty we assume without loss of generality that s' tkp s te, t tm. t' tn for change points t < to < tm < tn of 6. For the multi-index set A == A(k, n), define the cover {Ka) by K. ACo a A(e, m) A and K, 0 otherwise and the cover (K)) by K" = AC.
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for all A. Then we have Ka S Ky for all A and recover the MCbiF K UDEA D Ka and the nerve-based MCbiF K = N' JoE)) and similarly we recover K' and K'. For any k E IN and ag..... 0g E A. the intersections N KKa, are intersections of solid simplices and thus either empty or contractible. For a detailed proof of this fact, see the proof of Proposition 30 in Schindler & Barahona (202)). Lemma 33 now yields homotopy equivalences K + |K] and K' + | that commute with the canonical inclusions K] + [K*] and K - K", This shows the
equivalence between the MPH of the MCbiF and the MPH of the nerve-based MCbiF.
IT
Next, we prove Proposition about the dimension of the nerve-based MCbiF.
Proof of Proposition 16 Statement (i) follows directly from the definition in Eq. equation [ We show statement (ii) by induction, Base case: From the definition of the nerve-based MCbiF, it follows directly that dim No = 0 because the indices in A(m,m) correspond to mutu- ally exclusive clusters. Induction step: Let us assume that dim Nomtttn = n, then there ex- ist Co. C, E 0(1m))m+tn)) such that Co . O Cn # 0. As the clusters in partition (lmm+n+1) cover the set X there exist a cluster C E f(1m+n+1) such that C' n...n Cn * V. Hence, dim Nimmmmtn > + 1. If dim Aifm,tmtt C" E 0(mm+n+1) with C' n'nn Co n n On # 0 but C" n C * 0 contradicts that clusters of ((1m+++1) are mutually exclusive. Hence, dim fm n + 1, proving statement (ii) by induction.
n + there would exist a second cluster
We provide proof for the connection between Sankey diagrams and the herve-based MCbiF.
Proof of Proposition 17 The Sankey diagram graph S(0) = (V = V1 W VMM E = E
w w
11 is strict I-dimensional subcomplex of K = f because K kmmmm Vm C K and
Ktmafm+1 Em S K. This also shows that the zigzag filtration equation same vertices (O-simplices) and edges (1-simplices) as S(0).
5 contains exactly the
We next prove Proposition that characterises conflicts that can arise in a single layer of the Sankey
diagram.
Proof of Proposition
18 (i) Suppose that has O-conflict in [Im 1m+1). Then (1m)) ) $
0l(m+1)
and ((tm) I ((tm+1). This means that there exists C' E 0(r1) such that 3C" C" E 0(r2) with
CnC + 0, CCCCC C" * 0 and C' C" = 0, otherwise ((r)) S ((r2). Hence, 0(ri) L ( is
equivalent to Eu E Vm (the node corresponding to cluster C)) with degree deg(u) 2 2 in Em. An analogous argument shows that ((r:) z 0(r2) is equivalent to Eu E Vn++1 with deg(u) 2 2. This proves the statement.
(ii) Let x. U, E X form a triangle O-conflict for the interval [lm; 4m-), i.e., x ~tn y/ tt.. but otmt1 y pt.... In particular, the elements a. y, z are mutually distinct. This means there exist C1, C2,C2 E O(tm) and C₁ C22 E ((tm+1) such that æ, y E C1, 2 E C2. y. E C² and x E C₁. This is equivalent to C1 L C2 0, ni 0 and C1 n C₁ o 0, Cin # 0, C2 C & 0. Let u, u' E Vm correspond to C1 and Cz. respectively, and v, v E V correspond to C₁ and C₁ respectively. Then the above is equivalent to [u', v]. [v, ul, [u, o'I E Em,, which is again equivalent to the existence of path in E,, that has length at least 3.
(iii) The statement follows from the fact that every cycle in Em is even because the graph (Vm w Vm+1, Fm) is bipartite and the fact that every cycle in Em = Km is non-bounding because dim K Kmmdm+1 1.
B.3 PROOFS AND DETAILS FOR SECTION
We continue by proving that O-conflicts can induce violations of the strong triangle inequality as stated in Proposition]] 21
Proof of Proposition 27 Let x, W. E X lead to a triangle O-conffict for the interval [s, t). i.c.,
Br.r2 2 E [8, 0]: x on y ~rg and Ar f]: de y ve, 2. This means Do (x. < ri
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and (2, I r2- Let us define rs (22) ta-- We know that F3 # 11 and r's F 72 as otherwise I May y mry 2 for r3 E [s. tl, contradicting the lack transitivity. Hence, with- out loss of generality, min(Do,, v). Do,, (V), which is a violation of the strong triangle inequality.
< 72 rg. Then the above is equivalent to Dz, r3 2
Next, we state Proposition 35 that establishes the connection between the MPH of the MCbiF and that of the Merge-Rips bifiltration constructed from the matrix of merge times, Dø.s-
Proposition 35. Let us define the Merge-Rips bifiltration L bused on Do.s
(8) Then the O-dimensional MPH of the Merge-Rips bifiltration, L, and of the MCbiF, M, are equiv- alent, but the 1-dimensional MPH of L and M are generally not equivalent. Furthermore, if is strictly hierarchical, then C has a trivial I-dimensional MPH.
C (L* where - -o Vx,yEo: Do,( S
Proof of Proposition 35 First note that L Sugt is indeed a well-defined bifiltration be- i 2 s and & < 8' In partieular, L is also defined uniquely on the finite grid (a]] s S 1) with partial order (s, t) S (8', (') if 2 s', U. t'.
cause L C L
P= [(,,)) E (x)..,,
The proof of the proposition then follows from a simple extension of Proposition 36 in Schiinlerr & Barahooaa (2025) to the 2-parameter case. To see that the O-dimensional MPH of C and M are equivalent, note that both bifiltrations have the same 1-skeleton. Moreover, the I-dimensional MPH is generally not equivalent because L is Rips-based bifiltration and thus 2-determined, whereas M not 2-determined.
If A is strictly hierarchical, then Dø.. fulfils the strong-triangle inequality, and thus the Rips-based bifiltration leads to a trivial I-dimensional homology, see (Schindler Barahona 2025 Corollary 37). Hence, the 1-dimensional MPH of L is trivial.
Finally, we provide a brief proof for Proposition 22 linking the O-dimensional Hilbert function of pair of partitions and the graph Laplacian built from the conditional entropy matrix between both partitions.
Proof of Proposition 22 based MCbiF. Note that the graph G := Pt i tg/4 has the same vertices and edges as the simplicial
Using Proposition 17 we prove the statement with the equivalent nerve- Pr
complex kh,,2, which is I-dimensional and thus also graph according Proposition 16 This shows that HF (1,, (2) is given by the number of connected components in G. Furthermore, observe that f/ f = 1, and that the resulting matrix 7. is the Laplacian of the undirected graph G. Hence, dim(ker L) is equal the number of connected graph components (Chung 1997). proving the statement.
B.4 PROOFS AND DETAILS FOR SECTION 6
follows from the construction of MCbiF that the Hilbert functions are invariant to certain swaps of partitions in 0.
Proposition 36. HFF(s,() is invariant to swaps of partitions in sequence 0 between s and for t St
Proof. Let denote the change points of by 11 < < ... tM. Without loss of generality, bm and 1 bm+ for + S M. Let now consider permutation T [1, ..., .M + [l...., ......M. M] .......,]]]
for 1 < m and m + n < S M and define the permuted sequence of partitions 0, as 0, (tm) (f=(m))) Despite the permutation still get the same MCbiF for 0 and 9, for parameters < because
such that (i)
U U "Eršt Cea(r)
AC sErst (EE+(r)
U
U
This implies that HF& (s, t) is the same for 0 and 0,.
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C ADDITIONAL EXAMPLES
Our first example corresponds to the sequence of partitions analysed Fig
Example 37 (3-element example). Let X = [2, and we define ((0) = Ó, 0(1) =
(æ11 {a 8(2) {{a1), {z2, v3th. 0(3) = (s) { and 0(4) so that coarse- coarse-grainine graining with M = 5 change points. This example corresponds to Fig. Lt. and the and I-dimensional Hilbert functions are provided in Fig. 2
Note that (1, 2) < 10(2) and (2.3) < 10(3) indicates the presence of two triangle O- , (1,2) HF (2,3) = 0. See Fig. 7 for an illustra- tion. As shown in Proposition] [1 the triangle O-conflicts violate the strong triangle inequality of the matrix of first merge times Da 6E. e.g, Do(II, x3) 3 (axx)Do(T1x2)) Do (2(22,20)) va))
conflicts, which are not 1-conflicts because HF,
indicates the presence of a 1-conflict that arises from the higher-order inconsistencies of cluster assignments across partitions 0(1), ((2) and 0(3). See Fig. 7b for illustration. In particular, the equivalence relations #1 ~-1 22. 22 <2 23 and 23 ~3 I1 induce a 1- cycle [,1, 22 + [222, 333] + [x3, E Z1 (K1.3) and due to the lack of transitivity
In addition, 31)
the interval [1,3]. the I-cycle is also non-bounding, yielding a I-conftict. The I-conflict then gets resolved at 1 = because (4) = restores iransitivity on the interval [1, 4). See Fig. 7 for an illustration.
(a) connected 0-conflict
(b) 1-conflict
(c) no conflict
1F1
2
222
2
22
2
1'3
T2
T3
T'2
33
27:3
Figure 7: (a) Illustration of triangle O-conflict that violates the strong triangle inequality of the matrix of first merge times Dg.s (Fq (b) 1-conflict and (c) three elements that are no conflict 4, the confficts depicted here
due to global transitivity. If we choose correspond to the conflicts Example 1, r2 23 3 and
HFo(s.t) t)
(5, t) 1 2
0 1 2.25
0
2
2
4
4
6
6-
8
t
t
8(0)
6(3)
8(3)
8(4)
((5)
9(6)
8(7)
9(8)
6(9)
Figure 8: Hilbert functions and optimised Sankey diagram for the sequence of partitions 0 defined in Example 38
Example 38 (4-element example). We now consider the more complex case of a 4-element set
X {21, I2, Fz, v4). Let us start with 6(0) = Ô and append in sequence the 6 distinct partitions that contain two singletons and one cluster of size 2. i.e.. 0(1) [(1,, x2 {Is), {x1)), 8(2)
((x1), ), {Z2, vah. (Ia)). 8(3) = {(x]], J {x2), {wg, x4,,, ((4) (a {x2). {aa)h, ((5) =
(111 24), (x2), (2a)] andd (m1), (22, ). (xa) }} Finally, we append consecutively three
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partitions, each of which contains cluster of size 3, f.e. 8(7) = {[11 Fz, rz). ; 8(8) ((1), {æ2, 33, xz4) and ((9) = {1 23, T4 {22)}. 0 is a coarse-grainine non non-hierarchical sequence with M = 10 change points. See Fig. 8 for a Sankey diagram of 8.
To analyse the topological autocorrelation of 9. we compute the MCbiF Hilbert functions HF for dimensions k = 0, (see Fig. (.. We observe that HF
3) 1 min,c() [r(r) for all 0 I $ I 8, which implies that the hierarchy of B is broken after no-less than three steps in the sequence when starting scale s. Moreover, we can detect that 0 non-nested and has
higher-orler cluster inconsistencies because 1-conflicts emerge al scales 1 - 4, 5, 6, as indicated by non-zero values in HF1.
1. The I-conflicts get resolved one-by-on by-one through the partitions that contain clusters of size 3, and at t = 9, when the third such partition appears in 8, all 1-conflicts are resolved.
Finally, we show example that demonstrates how conditional entropy does not detect 1-conflicts in general.
Example 39 (CE, cannot detect I I-conflicts). Let X = {x), 32, I3, T4) for which we consider two
different sequences of partitions O(t) and (tt such that ((1) (11) x2), {2z}. {x4}},
0(2) = (22) = (111). {22, F3). {2a}} but 0(3)
(1 {2a}} # 0(3)
{ ) { (x)) See Fig. for Sankey diagram representation of the two sequences of partitions. Note that and n only differ at scale = 3. However, this difference is crucial because a I-conflict emerges in 0 at scale = whereas y has only triangle O-conflicts and no I-conflict. Note that 0 corresponds to the toy example in Fig. 7 with one additional isolated element.
(a) Sequence 0 (1 (1-conflict)
(b) Sequence y (no 1-conflict)
{4
{
{
{22}
(x)
(as)
(ur)
{as
(rr)
(daa)
(ma)
{xH
{144)
8(1)
8(2)
0(3)
v11)
1/(2)
17(3
Figure 9: Sankey diagrams for sequences 0 and 17 defined in Example 39 Note that a 1-conflict emerges in 0 at scale = 3, but has no 1-conflict.
In accordance with our theoretical results developed in Section 3.1 we can use the 1-dimensional Hilbert function HF, to detect the I-conffict in @ and distinguish the two sequences. In particular, HF, (0(1), = but HF, (nj(i), = 0 for all i,j E [1. 2, 3). i S j. In contrast, the conditional entropy H (see Eq. (6) cannot distinguish between the two sequences as they yield the same pairwise conditional entropies. In particular, H(B(i))()) = H(m(1) In(i) = log 2 for o j. This demonstrates that the conditional entropy cannot detect higher-order cluster inconsistencies in sequences of partitions.
D DETAILS ON EXPERIMENTS
D.1 REGRESSION TASK
We first provide a rigorous definition of the space of coarse-graining sequences of partitions.
Definition 40 (Space of coarse-graining sequences of partitions). The space of coarse- course-grainig se- quences of partitions, denoted 11 is defined as the set of coarse-graining sequences 0: [0, xc) +
Ix with |X| = N and M change points imm = O..... M 1, such that (0(s)] Z |#(t) I, Vs < t,
which start with the finest partition O(t = =) Ô and end with the full set A(tas M 1) =
For our experiments we sample randomly from the spaces 1130 and 1120 Note that [ is given by the exponentially growing Bell numbers By with B10 = 115, 975 > B5 = 52 Stanley (201)).
Figure 10 shows the correlation between the minimal crossing number y R=()) (Eq. 11 and summary statistics of the five feature maps under investigation: the consensus indices VI. ARI and MOD, and the MCbiF topological average measures En and č In addition to the results already described in the main text, we also observe that the correlation between VT and op (r = -0.32 for
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N 5 -0.48 for N 10) is stronger than with é1 (r -0.12 for N 5, -0.34 for N = 10). This can be explained by the fact that VI and čσ can both be computed from pairwise observe strong correlation between é and či (r = 0.52 for N and r 0.43 for = 10) because of the dependencies between 0- and 1-conflicts, see Section 3.1
interactions of clusters in contrast to C₁. see Section Furthermore,
5
N 10
027
D22
50
TWOIE
10
300
Pearson correlation (r) between crossing number consensus indices VI, ARI and MOD, and MChiF-based conflict measures Op and 6j for N and N 10.
Figure 10:
Note that we can consider our five feature maps as M x M greyscale images, where HF HF1, ARI and MOD are symmetric and CE is asymmetric. The raw label encoding of A is also similarly interpreted as an N M greyscale image. For our regression task, we train a simple CNN (LeCun) & Bengio, 1998) with one convolution and max-pool layer and one fully connected layer and also simple MLP (Bishop 2006) on the flattened images with one or two hidden layers and dropout (Sri) vastava et al. 2014). For each feature map (or their combinations) separately, we perform hyperpa- rameter optimisation for the number filters (ranging from to 6) and kernel size (chosen as 4, 8, 16, 32 or 64) in the CNN and the number of nodes (chosen 4, 8, 16, 32, 64, 128 or 256), number of layers (1 or 2) and dropout rate (chosen as 0.00, 0.25 or 0.50) in the MLP. We use the Adam optimiser (Kingma & Bal 2017) with batch size 32 and learning rate chosen 0.01, 0.005, 0.001, 0.0005 or 0.0001 for training. We train both MLP and CNN over 150 epochs with early-stopping and patience of 10 epochs.
As an additional baseline model, we train a graph convolutional network (GCN) (Kipf & Welling 2017) with a regression head on the weighted adjacency matrices of the Sankey diagram graph S()) (Eq. 9.. where the edges are weighted according to the number of elements in the overlap of two clusters. We choose three-dimensional node features consisting of a constant, the normalised weighted degree and the layer number of the node in the Sankey diagram. We perform hyperparam- eter optimisation for the number of hidden dimensions (16, 32, 64, 128), number of layers (1, 2 or 3) and dropout rate (0, 0.2 or 0.5). We use again Adam optimiser with batch size 32, learning rate chosen as 0.01, 0.005, 0.001, 0.0005 or 0.0001 and weight decay chosen as 0.0 or 0.0001, as well as the Reduce 1 lateau learning rate scheduler and train the GCN for 150 epochs with early-stopping and a patience 10 epochs, consistent with the other models.
We perform full grid search of the hyperparameter space for MLP and CNN trained on the differ- ent feature maps (or their combinations). We perform 145 trials hyperparameter search with the Tree-Structured Parzen Estimator (TPE) (Watanabe 2025) for the GCN,
full grid search was prohibited by increased computational complexity. We used the train split of our data for training and the validation split for evaluation and hyperparameter selection. Below, we detail the hyperpa- rameters for the best models trained on the different features, which were chosen according to the performance on the validation split. We first report details for N 5:
Optimal model for raw label encoding at N 5: CNN with 16 filters, kernel size 4 and learning rate 0.001.
Optimal model for Sankey graph S(0) N 5: GCN with three hidden layers of dimen- sion 128 each, no dropout, no weight decay and learning rate 0.01.
htttps::/docs.pytoorrrr schedu 1er het.eee.eeddueeeeoaaataete http://w.w..oooo.o.......aaaaaaa
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Table Regression task. Test R2 score with bootstrapped 95% confidence intervals LRR CNN, MLP and GCN models trained different feature sets for N =
0.413 (0. 11.388-- (1.38-8--330
0.354 (0.3 00.277 0220800))
0.351 (0. 00.325 00.3225-0333 01375)
0. 0.345 (0 3.0 0..39)
0.246 00.221 (1.26)
0.3 0.312 (0.286600333)
MOD
1,166 (0.1 (0..142-0..0000
08888888
0.422 (0.39- (0.397-0.446) 00446)
0.214 191 -0,236)
0. 0.246 (0 22) 1721--0.2700
29244 (0.268 0.3999
0.256 23030 0.280)
ARI
0.392 (0.367-0-114)
0.492 0.492 (0.468- 0.514)
0.409 (0.385 0.432)
0.47 (0.434 0.449
(0.431-0,476)
0.45% (0.435 0.400)
CE
0.539 (0:518-0.559)
0.544 00.522-0- 0.565)
0.54 (0.52000..00. 0.5611
0.516 (0 (0.40440.537)
0 (0-486-0-522 0-527)
0.514 (0.493 0.534)
HF, & HI,,
0.48 0.486 (0.463-0 509)
0.504 (0 (.566))
0.49 (0.469-0.55) 5153)
0. 0.448 (0.425-0 470)
0.448 10-42-00--0021 470)
0.450 0,450 (0.426 0.413)
HP,
0. 147 (0. 12512.200111 168)
0.155 (0.133--))) (0.133-00.16) 0..16))
1.0 (0) (30 (30-0.170)) 170)
0,214 (0.19) 19119-2--333
0,211 (0 (0.188-00222 23-
0,212 (0.191 0.232)
HFP
0.416 (0. 392-0-42443 0-438
0,229 (0.20 11200-0.200 0.245)
Sunkey grupls
0,078 (0.0000000095)
0:267 (0.241-0) (0224-00299) 0.293)
0.104 104 (0 124)
0.072 0,072 (0 0.092)
0.036 (0.019 0.053)
0.038 022-0002000
Row label encoding
Method
LR CNN
MLP
GON
IRR CNN
MLP
GCN
and N = 10.
N
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Table 4: Train R scores of LR, CNN, MLP and GCN models trained on different features for N and N 10.
Raw label encoding
Sankey graph
N
Method
HFP
HF1
HF, & HFF,
CE
ARI
MOD
LR
0.096
0.163
0.493
0,550
0.409
0.194
0.419
5
CNN
0.321
0.170
0.509
0.562
0.515
0.464
0.460
MLP
0.254
0.160
0.499
0.547
0.439
0.366
0.396
GCN
0.397
LR
0.061
0,230
0.456
0.522
0.464
0.255
0.370
10
CNN
0.112
0.220
0.456
0,519
0.476
0.376
0.368
MLP
0.114
0.218
0.453
0.515
0.468
0.368
0.282
GCN
0.234
Optimal model for HFo & HF N 5: CNN with filters, kernel size 3, and learning rate 0.001.
Optimal model for CE at N = 5: CNN with 8 filters, kernel size 2, and learning rate 0.005.
Optimal model for ARI at N 5: CNN with 8 filters, kernel size 2, and learning rate 0.005.
Optimal model for MOD at N = 5: LR.
We next report the details for N = 10:
Optimal model for raw label encoding at N = 10: CNN with 8 filters, kernel size 3 and learning rate 0.01.
Optimal model for Sankey graph S(0) at N 10: GCN with three hidden layers
mension 128 each, dropout rate 0.25, no weight decay and learning rate 0.005.
Optimal model for HF, & HF at N 10: LR.
Optimal model for CE at N = 10: MLP with a single layer of 256 nodes, no dropout and learning rate of 0.001.
Optimal model for ART N 10: CNN with filters, kernel size 2, and learning rate 0.005.
Optimal model for MOD N 10: LR.
We present the train RD scores for the optimised LR, CNN, MLP and GCN models trained on the different features in Table [ The test Rt2 scores are presented in Table [ in the main text. We also report 95% confidence intervals for the test Ri2 score in Table 3 which computed using bootstrapping with 5,000 iterations on the test data.
D.2 CLASSIFICATION TASK
We first provide the definition for order-preserving sequences of partitions.
Definition 41 (Order-preserving sequence of partitions). When partition ((tm) is equipped with total order <m on the clusters it is called an ordered partition F Such partition induces total preorder S. X (Stanley 2011), i.e., if [x); <m [y]t then Sm y. We call the 0 order-preserving if there exist total orders (51...,, <M) such that the total preorders (SI...,,Ms) are compatible across the sequence, i.e., Vl, m we have Se y iff x S.m y. Vx. E X.
According to this definition, a sequence non non-order-ppp penderrpreeevvigg there is no total order on X that is consistent with all the total preorders induced by the partitions f(t).
Thhe ranking Tm Vn + {1...., Vnl) I of the vertices V., in the Sankey diagram S(0) is one example of total order Sm the clusters, Section F.
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500
400
Frequency
500
200
100
Total number of swaps in
Classification task: Histogram of the number of swaps in non-order-pressering sequences 0 (class y 1). See text for the scheme introduce random swaps in the cluster assignments means to break ordee-preservation.
Figure 11:
Details on Synthetic Data. We generate order-preserving (y = 0) sequences E I through the following scheme: Let assume that have total order X {x), ....,....................... IN given by the element labels, i.c. 22 I, if i < j. We construct each (. m 0, .......4 M - 1. by cutting x into clusters of the form C {I, ....t........................... #iti........,
easy verify that 0 is indeed order- preserving. We adapt this scheme to generate sequences 0 E IT that are non-order-pres -rrrorrrrrrrr-iiinggg (y 0): Again, we start by constructing each sequence (t... through cutting the ordered before. Additionally, with probability arbitrary elements x, y E X. and are large enough, the so-generated sequence 0 is almost surely non-order-preserving. We chose N 500 and M 30 to demonstrate the scalability of the MCbiF method.
as 0.1, we swap the cluster assignments in (tt... ) for two
The number of clusters of all our generated sequences of partitions 0 II for both classes is decreasing linearly. Moreover, the average number of swaps for sequences with = is 2.98 for choice of p 0.1, see Fig. 11
Results. We find significant difference between the baseline consensus indices VI, ARI and MOD of order- pedee-preserving. (y = 0) and non-order-preserte (y = 1) sequences. In contrast, we observe statistically significant increase of Eo and for order-preserving sequences (Fig. 12
==0
y=
0.81
0.28
6.00
0.38
0.94
003
0.81
0.27
8
0.27
0.37
0.93
& 0.81
4.00
ARI
MOD
0.27
0.36
0.93
0.80
2.00
0.27
0.35
5
0.92
0.80
0.00
0.26
Figure 12: Difference between the baseline consensus indices VI, ARI and MOD and the MCbiF topological average measures C and e of order-preserving (y = 0) and non-order-preeservnn (y 1) sequences of partitions (**** indicates 0.0001, Mann-Whitner Whitney U test).
We measured the performance variability in the classification task with bootstrapped 95% confidence intervals, see Table 5 which were computed using bootstrapping with 5,000 iterations on the test data.
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Table 5: Classification task. Test accuracy with bootstrapped 95% confidence intervals of logistic regression trained on different features.
Row lahel
encoding
HFo
HF,
CE
ARI
MOD
0.53 (0.50-0.57)
0.56 (0.53-0.60)
0.97 (0.96-0.98)
0.50 (0:47-0.54)
0.49 (0.45-0:53)
0.46 (0.43-0.50)
D.3 APPLICATION TO REAL-WORIL TEMPORAL. DATA
Data Preprocessing. The temporal sequences of partitions computed by Bovet et al. (2022) available at: https: /ataveeree..eeeeeeeeee ha rvard. eew.filedd/file... xhtml?fflle 1 5 We restricted the partitions to the N 281 mice that were present throughout the full study period ensure well-defined sequences of partitions, and considered the first nine temporal resolution val-
6692
ues Ti, = .,... 9, since O. is an outlier. Note that the sequences tend to be fine-graining, see Fig.
13
30
Temporal resolution
==15
ry =
ti
Ts 5-1h
15
==5
==8h
10
:s == 24
7=
5
1
2
5
8
Scale weeks]
Figure 13: Number of clusters weeks for different temporal resolutions where larger values of r produce higher number of clusters because of the increased temporal resolution.
10
| HF [HF HF, 1z
Hilbert distance
6-
4-
10-1
100
10a
102
103
10a
10a
10c
T [in s]
T2
To
Figure 14: Hilbert distance between forward and backward sequences of and 6 for different tem- poral resolutions 7.
Time Reversibility. In the main text, we restricted our analysis to the so-called forward Flow Stability sequences of partitions. However, by reversing time direction, Bovet et al. (2022) computed a second set of backward sequences. For each temporal resolution т,, we thus get a forward and backward sequence denoted by of, and 8%.- respectively. Here forward and backward sequences partitions for different T;, and we compute the Hilbert distance (x(((( F (8t,) 2|2 for &o = 0.
the MCbiF to compare the
Fig. 14
We observe that the Hilbert distance between forward and backward sequences is high for T2 because the large-scale group structure changes significantly the study period, so that the temporal
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flows low resolution 72 not reversible. In contrast, the Hilbert distance between forward and backward sequences is lower for тy because the underlying social groups are more stable over the study period, leading to increased time reversibility at the high temporal resolution. However, 74 leads to the lowest Hilbert distance between forward and backward sequences, showing that the temporal resolution not only leads the most hierarchical forward sequence but also temporal flows that are most time reversible.
E BACKGROUND
E.1 SANKEY DIAGRAMS
Non-hierarchical sequences partitionns are visualised by M-layered flow graphs S(0) (V
Vi U... W VM E E1 ... w 1M-1) called Sankey diagrams (Sankey 1898 Zarate al. 2018 where each level m = 1..... . .M corresponds to a partition and vertices Vn represent its clusters while the directed edges Fm between levels indicate the overlap between clusters:
Vm == {(m, i) 1 i S [0(tm)]): Em [[(m, i), (m + 1,5)] ((tm); ((mm+1)) # 0). (9)
where [u. F. denotes directed edge from u E Vn to E V If 0 is hierarchical, the Sankey diagram S()) directed tree-a merge-tree if coarse-graining or split-tree if fine-graining. The graph S(0) is also called an alluvial diagram (Rosvall & Bergstrom, 2010).
Sankey diagrams are studied in computer graphics as they allow for the visualisation of complex relational data. In this context, Sankey diagram is represented as layout on the plane, whereby the nodes in each layer Vm are vertically ordered according to ranking Tm Vm - (I..... [m and the layout of the Sankey diagram is then defined by the collection of such rankings, (TI...., TM.) For visualisation purposes, the layered layout should ideally minimise the number of crossings between consecutive layers, where crossing between edges (u, v),, [u'; o') F.m OC- curs Tm (u) > Tm (u') and (v) < Tm++1 (v') or vice versa, and the crossing number (Warfield) 1977 is given by:
M-1
KO(T) = Σ
Σ
(10)
where denotes the indicator function. The crossing number ko(T) of the layout of the Sankey diagram S(0) can be minimised by permuting the rankings in the layers, and we denote the minimum crossing number for the layout as:
Kg = min (T))).
(11)
This problem known to be NP-complete (Garey & Johnson 2006) and finding efficient optimisa- tion algorithms is an active research area Zarate et al. 2018 Li et al. 2025
E.2 MULTIPARAMETER PERSISTENT HOMOLOGY
Multiparameter persistent homology (MPH) is an extension of standard persistent homology to n > parameters. first introduced by Carlsson & Zomorodian (2009). We present here basic definitions,
see Carlsson & Zomorodian 2009);; Carlsssn et al. (2009) Botnan & Lesnick (202) for details.
Simplicial Complex, K be a simplicial complex defined for the set X, such that K C 2 and K for V 7 C a E K. The elements of K are called simplices and k-dimensional simplex (or k-simplex) can be represented as = [I,,,.., where X1, Fk+1 E X and we have fixed 1 to edges, and k 2 to triangles. We define the k-skeleton K₃ of K as the union of its n-simplices for I k. We also define dim K) as the largest dimension of any simplex K.
an arbitrary order on X. Note that k 0 corresponds
vertices,
Multiparameter Filtration. Let define the parameter space (P, <) the product of 2 partially ordered sets P == P1 x ... x Pn. i.c. a < for a, b E P and only if a; < b, in P. for i -,....P n. A collection of subcomplexes (Ka) aeea with K Unero Ka and inclusion
(Ka)aeRa
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maps (ia.b: : K Kb) Jasb that yield commutative diagram is called a multiparameter filtration
(or bifiltration for n = 2). We denote by birth(or) S P the set of parameters, called multigrades (or bigrades for n 2), at which simplex E K first appears in the filtration. For example, the filtaation K = {a E K f(o) S <a) for a filtration function f K + maps each sublevel filtration
simplex unique multigrade f(o), i.e., [birth(o)] = 1. A filtration is called one-critical if is isomorphic to a sublevel filtration, and multi-critical otherwise.
Multiparameter Persistent Homology. Let H₃ for k E 10..... dim(()))))))) denote the k- dimensional homology funcior with coefficients a field (Hatcher, 2002), see Appendix EE.3 for details. Then Hy applied the multiparameter filtration leads to mulliparammeter persistence perriaasteeeeppsssttetnene mod- ule, i.e., a collection of vector spaces (H& )aER". which are the homology groups whose ele- ments are the generators of ks-dimensional non-bounding cycles, and linear maps (ta,b b = Hu (Ha.)) Hx(KK)) + Hx(K+)) Jasb Jasb that yield commutative diagram called multiparameter persistent ho- mology (MPH). For dimension & = O, H₂ captures the number of disconnected components and for the number of holes. Note that, for
we recover standard persistent homology (PH)
Edelsbrunner et al. 2002).
Hilbert Function. While barcodes are complete invariants of 1-parameter PH 1), the more complicated algebraic structure of MPH (n 2 2) does not allow for such simple invariants in gen- eral; hence, various non-com non-complete invariants of the MPH are used in practice. We focus on the k-dimensional Hilbert function (Harrington et al. 2019 Botnan & Lesnick 2023 defined as
(12) which maps each filtration index a to the k-dimensional Betti number the corresponding complex K We call the k-dimensional MPH trivial if HF = O. The Hilbert distance is then defined as the L2 norm on the space Hilbert functions and can be used to compare MPH modules.
HF P - No, a rank (va.a)) dim dim(Hx&(' (K*))"))
E.3 THE HOMOLOGY FUNCTOR
We provide additional background on simplicial homology and its functoriality, following Hatcher 2002).
Simplicial Homology. Let K be simplicial complex defined on the finite set X. For a fixed field (the RIVET software uses the finite filed k = Z2 (Wright & Zheng 2020)) and for all dimensions k E {0, 1,.., dim(K) we define the k-vector space C2 (K) whose elements given by formal sum
Σ 0,
(13)
dim(o)=k
with coefficients 0g k called a k-chain. Note that the k-dimensional simplices o (o, 12,,,... ØR] E K form basis of Ck(K). For a fixed total order on X, the boundary opera- for is the linear map 0n Ch - Ck-1 defined through alternating sum operation the basis vectors o = [Do, æ1, ..., Øk] given by
0x(d)= = (-11) F1, 2i,..., xx].
where di means that vertex n; is deleted from the simplex a. The boundary operator fulfils the property im 0k+1 C. ker Oz. Hence, connects the vector spaces Ck. k {0, 1, ..., dim(K)) through linear maps
8 8k+1 C Se, Ch-1
82 C1 2 Co 3.
0.
leading to sequence of vector spaces called chain complex. The elements in Z := ker Ôr are called k-cycles and the elements in B, == 0.41 are called k-boundaries. Finally, the k--th homology group H2 is defined the quotient of vector spaces
Ha Zz/BR.
(14)
whose elements are equivalence classes [2] of k-cycles 2 E Zg. Each equivalence class [z] * corresponds generator of non-bonn cycles, i.c., k-cycles that not the k-boundaries of + 1-dimensional simplices. This captures connected components at dimension = 0, holes at
k and voids at k = 2.
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Functoriality of Hg. For fixed k. H2 can be considered as a functor H₃ Top - Vect, where Top denotes the category of topological spaces whose morphisms are continuous maps and Vect the category of vector spaces whose morphisms are linear maps. In particular, each topological space K is sent to a vector space (k) and a continuous map g : K + K" is sent to a linear map Hx(gg) H& (K) -> Hg(K") such that compositions of morphisms are preserved, i.e., He(g f)
(g) Hf) for two continuous maps f and
E.4 ZIGZAG PERSISTENCE
We provide background on zigzag persistence, which was first introduced by Carlsson de Silva 2010). For additional details, see Dey & Wang (2022).
Zigzag Filtration. Let Km be a simplicial complex defined on the set X for m 1,.... M. either km C Km+1 or Km+ S CK", for all m ....., M, we call the following diagram zigzag filtration:
*² . KM- K
where Km ++ Km is either a forward inclusion KT > Km+1 or backward inclusion Km Km+1, While forward inclusion corresponds simplex addition, backward inclusion can be interpreted as simplex delction.
Zigzag Persistence. Applying the homology functor H3 to the zigzag filtration leads to a so called zigzag persistence module given by:
Hx(K²) ++ (2(K2) +..... + Ha(KM-1) ++ (K(KM),
where (*(K")) + H₂ (k is either forward backward linear map. Using quiver theory, it can be shown that a zigzag persistence module has a unique interval decomposition that provides a barcode as simple invariant.
F DETAILS ON INFORMATION-BASED BASELINE METHODS
Information -base measures can be used to compare arbitrary pairs of partitions in the sequence (Meilä, 2007). Assuming uniform distribution on X, the conditional probability distribution of ((1) = (CI.... } given 0(s) (Ci,..., Cm3 is:
Pajl
(15)
ICyl
and the joint probability Ps. [. is defined similarly. The conditional entropy (CE) H(t[s) is then given by the expected Shannon information:
[oce)) [0(+)]
H((ss) = Σ Σ
Pa.lis log( (1)))))
(16)
1=1
It measures how much information about 0(t) we gain by knowing 0(s). If 0(s) S 0(1) there is no information gain and H({[s) To summarise the pairwise conditional entropies in the sequence 0, we define the M x M conditional entropy matrix CE by:
CE. (t,((t))
(17)
for i,j f1,...,MM M). Furthermore, we can compute the variation of information (VI) vi(s,t) =
H(s)t) + H(t]s), which is metric. Both CE and VI are bounded by log N.
Extending information-base measures for the analysis and comparison of more than two parti- tions is non-trivial. However, the pairwise comparisons can be summarised with the consensus index (Vinh et al. 2010) which can be computed as the average VI:
Emi VI(t,,tzz
VI(0) = M(M - 1)/2
(18)
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Reviews:
[{"review_id": "VL4wE9kYLX", "reviewer": "Reviewer_NmJi", "summary": "This paper introduces the Multiscale Clustering Bifiltration (MCbiF), a topological framework for analyzing multi-resolution, non-hierarchical clusterings that describe data at different levels of coarseness. MCbiF models how clusters intersect and evolve across scales using a two-parameter filtration of simplicial complexes, whose structure is studied through multiparameter persistent homology. The resulting stable Hilbert functions capture key topological properties: dimension-0 features measure the nestedness of clusters, while dimension-1 features quantify higher-order inconsistencies between them. The method generalizes dendrograms (for hierarchical cases) and extends Sankey diagrams to represent higher-order relationships.", "questions": "- Could the authors elaborate on how MCbiF could be applied to modern large-scale or high-dimensional datasets (e.g., image, text, or graph data)? This would help clarify the method's practical relevance beyond theoretical or synthetic settings.\n\n- Why were only information-theoretic measures (e.g., conditional entropy) used as baselines? Including comparisons with recent representation learning or topological deep learning methods (e.g., graph neural networks, persistence-based embeddings) could strengthen the empirical validation.\n\n- While the paper provides theoretical insight into Hilbert functions, could the authors give more intuitive examples or visualizations showing how specific topological patterns correspond to interpretable data behaviors?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper is well-grounded in theory and method, with clear mathematical development and interpretable ideas. Its main strength lies in the clarity and rigor of the framework, which extends topological concepts in a coherent way to analyze multiscale clustering.", "weaknesses": "However, the empirical evaluation is limited and lacks modern benchmarking. Experiments are mostly conducted on synthetic datasets and a single small-scale real-world example (wild mice social groups), which limits generalizability. The comparative analysis is restricted to information-theoretic baselines (conditional entropy), with no inclusion of state-of-the-art representation learning methods such as graph neural networks or topological embeddings. Performance variability and robustness analyses are not reported, and code availability is not mentioned, hindering reproducibility. While the theoretical development is strong, the translation to practical machine learning applications is limited, reducing the paper's accessibility and impact for a broader representation learning audience.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "xgmOVQVLxm", "reviewer": "Reviewer_tfaN", "summary": "The paper considers the following setting. Given is finite set $X$. For each real number $t \\geq 0$, there is a partitioning of $X$, in other words, we have a family of different clustering ways. Importantly, it is not assumed to be hierarchical. The paper proposes a way to use 2-parameter persistent homology to describe such a family. The construction is elegant: given values $s \\leq t$, one looks at all the clusters occurring between $s$ and $t$ and inserts a corresponding simplex into the complex $K_{s,t}$.\nThe paper proves algebraic properties of the resulting 2-persistence modules (pointwise finite-dimensional, finitely presentable and block-decomposable). Then the paper concentrates on the Hilbert functions (just pointwise dimensions) of the resulting bifiltrations and derives new measures of conflict and interpretations of the information captured by the TDA techniques. There are also experiments, offering new insights for, e.g., a real-world dataset showing how the grouping of mice changes over time.", "questions": "I wonder if every block-decomposable module can be obtained from some clustering by the proposed construction? Of course, the module must also be assumed to be zero outside of the $s \\leq t$ triangle.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The construction of bifiltration is really nice and elegant. While the formal definition might be hard to digest for non-experts, the illustration in Fig. 1 does a great job explaining what is really going on.\n- The paper provides new quantitative measures to clustering.\n- The results generalize the previously known Sankey diagrams.", "weaknesses": "- As the paper admits, Hilbert functions are a rather crude invariant that loses some information. The paper poses exploration of other invariants as future work, which is fair enough.\n- The first experiment (predicting crossing number) seems to be a bit weak, considering that even for MCBif the Pearson is still 0.544. I mean, the whole set up looks a bit artificial, but I am not an expert in clustering.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "axnzqws8FE", "reviewer": "Reviewer_5d9w", "summary": "This paper proposes a method to analyze the defect if hierarchically in a\nclustering using novel filtrations for multiparameter persistent homology.\n\nThe multipersistence module induced by this filtration is shown to satisfy\nseveral desirable properties, such as being decomposable in\nrectangles, or characterize defects in a nonhierarchical clustering.", "questions": "see weaknesses.\n - Are there some stability result w.r.t. $\\theta$?\n - Is there a link with the interlevelset filtration (which has similar\n guarantees)? I'm not sure this is possible since IIC, the Mayer-Vietoris\n sequence isn't valid for the non-nerve version, but this might be possible\n with the nerve version?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Detailed mathematical analysis. The proof seem sound.\n  - The resulting filtration is mathematically satisfying. \n  Rectangle decomposability a good to have.\n  - The analysis with the 0,1-conficts is also interesting", "weaknesses": "- Code not available.\n - Maybe a bit math-heavy for this conference. I also think a little drawing\n could help for some definition/result.\n - Several definitions / proofs are hard to follow.\n - Experiments.\n   - I'm not familiar with the state of the art, but from what I see, the only\n   competitor is the conditional entropy introduced in 2003. It is thus hard\n   to judge the practical impact of this construction.\n   - I guess that this filtration can become easily very large but $N,M$ feel\n   a bit small. How does this filtration scale w.r.t. these parameters? and\n   what about its nerve equivalent? \n\n\n\nminor comments.\n- l157. I think there is an issue with the indices. shouldn't it be\n$m,m,m+1,m+1$ ?\n- l173. Shouldn't the $P_i$s be totally ordered? \"operation of building\nsubsets\" this is a bit vague.\n- l216. Every multifiltration is multi-critical?\n- l248. Unless I missed something, Rivet can handle arbitrary module\npresentation, so MPH can be computed for arbitrary homological degree. Rivet is\nalso slightly outdated to compute the hilbert function. For instance, `mpfree`\ncan compute the betti numbers significantly faster (and for arbitrary degree as\nwell), for which there is a python interface in `multipers`. Furthermore, as\nthe induced module rectangle decomposable (Prop 21), the rectangle\ndecomposition (computable in `multipers`) should recover the module.\n- Def 7. Not very intuitive.\n- l269. What is a strictly hierarchical clustering? Is the relation strict on\nthe restriction on which it is non-constant?\n- l303. $C$\n- l333. $y$\n- Table 1. Who is raw $\\theta$, and what about corollary 15 instead of feeding\nit to a regresser?\n- prop 21. finitely presentable? and in the proof, I think it requires prop 23.\n- l701. since $\\theta$ is piecewise constant?\n- prop 23. I think this should be detailed a bit more.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)