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REVIEW 3 major objections 3 minor 90 references

Persistent Multiscale Density-based Clustering

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read PLSCAN replaces HDBSCAN*'s manually tuned minimum cluster size with a persistence-based automatic choice, reading every stable leaf-cluster threshold from a single condensed tree.

desk verdict The leaf-tree construction in PLSCAN does not reproduce HDBSCAN*'s leaves for unbalanced merges, so the paper's central equivalence claim is unsupported — though the code and benchmark are solid and worth refereeing. read the letter →

arxiv 2512.16558 v3 pith:2TNFU2J4 submitted 2025-12-18 cs.LG

Daniël Bot , Leland McInnes , Jan Aerts This is my paper · ORCID
classification cs.LG
keywords density-basedclusteringHDBSCAN*minimumclustersizepersistenthomologyscale-spaceleaftreeexploratorydataanalysishierarchical
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

PLSCAN is a density-based clustering method that removes HDBSCAN*'s main tuning parameter. HDBSCAN* needs a minimum cluster size m_c, and its output depends heavily on this value; PLSCAN instead builds a leaf tree that records, for every m_c, which local density maxima survive, then selects the m_c with the largest total persistence. The paper's key claim is that this can be done without rerunning HDBSCAN*: one condensed tree built at a low m_c already contains every merge that would occur at higher thresholds, with unchanged sizes and distances. On real-world benchmarks, PLSCAN with size persistence reports a higher average ARI than HDBSCAN* (0.72 vs 0.58 at k=4) and is less sensitive to the number of neighbours k. If the claim is right, exploratory users no longer have to guess a cluster-size threshold, and they gain a multiscale hierarchy of alternative stable clusterings.

What carries the argument

The leaf tree is the carrying object. It converts an HDBSCAN* condensed tree—a list of merges with distance d_mut and child size s—into birth-death intervals in minimum cluster size: s_max = min(left size, right size), s_min = max(children's s_min), with the parent's birth delayed until all downstream leaves vanish. The persistence trace sums s_max - s_min over all currently existing leaves and is evaluated only at the discrete s_min/s_max values where the leaf set changes. The formal counterpart is a metric d_s in which the distance between points is the smallest pruning parameter that puts them in the same leaf cluster, plus noise elements that record when a point ceases to belong to any l

What would settle it

Rerun HDBSCAN* on a small hand-built dataset with unbalanced nested clusters for every m_c from 2 to N, and compare the leaf clusters at each threshold with the intervals PLSCAN predicts from its single leaf tree; any mismatch in a birth or death threshold refutes the one-tree claim. A simpler check: compare merge sizes and distances in condensed trees computed at m_c=2 and at a large m_c on the same data—if they differ, the leaf tree foundation is violated.

Watch

Extended reading notes

Core claim

The central discovery is that the minimum cluster size in HDBSCAN* can be treated as a scale parameter, and the full family of HDBSCAN* leaf clusters across all m_c values can be recovered from one condensed tree. The paper introduces a leaf tree whose segments carry the interval (s_min, s_max] of thresholds for which they are leaf clusters: a child stops being a leaf at s_max when either child branch disappears, and a parent is born only after all its downstream leaves have disappeared, s_min = max(...). Summing the size persistence s_max - s_min over all leaves gives a persistence trace over m_c, and PLSCAN chooses the threshold with maximum total persistence; flat labels then follow in a

Load-bearing premise

The load-bearing premise is that a condensed tree built at a low minimum cluster size contains every merge HDBSCAN* would produce at higher thresholds, with the same merge sizes and distances; if pruning changes those quantities, the leaf tree misdescribes HDBSCAN*'s actual leaf clusters.

Editorial extensions

If this is right

  • HDBSCAN*'s minimum cluster size m_c no longer needs to be supplied by the user: PLSCAN returns the threshold with the maximum total persistence and the corresponding leaf clusters.
  • Because k and m_c are decoupled, PLSCAN's clusterings change predictably with the neighbour count k; the paper reports lower sensitivity than HDBSCAN* across perturbations of k.
  • Local maxima of the persistence trace provide alternative stable clusterings at different levels of detail, giving an analyst a multiscale view of the data rather than a single partition.
  • The persistent-homology equivalence means the leaf-lifetime construction is not specific to cluster size: it applies to any monotonic pruning parameter of a linkage tree.
  • On the evaluated datasets, average ARI at k=4 rises from 0.58 for HDBSCAN* to 0.72 for PLSCAN with size persistence, and low-dimensional runtimes are competitive with k-Means.

Reading between the lines

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

  • Editorial inference: the 'one tree contains all thresholds' idea should transfer to other monotonic pruning schemes, such as density persistence or excess-of-mass pruning, producing parameter-free hierarchies for those methods as well.
  • Editorial inference: the selection rule is a single straight cut through the leaf tree, so strongly unbalanced cluster sizes or two-sided structures may need multi-level cuts; the paper itself flags the E. coli and Yeast Galactose datasets as cases where this matters.
  • Editorial inference: the persistent-homology reformulation opens the door to using barcode stability theorems to bound how much PLSCAN's clusters change under resampling—a stronger guarantee than the reported k-sensitivity experiments.
  • Editorial inference: a synthetic-data study with known cluster sizes could separate two claims the paper runs together—that the leaf tree faithfully reproduces HDBSCAN* leaf clusters at all m_c, and that maximum total persistence is a good quality criterion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. PLSCAN replaces HDBSCAN*'s user-set minimum cluster size m_c with an automatic persistence-based cut. It constructs a condensed tree at a low m_c, converts it into a 'leaf tree' storing the m_c intervals (s_min, s_max] over which each cluster segment is a leaf, computes a persistence trace by summing leaf size persistences, and selects the m_c maximizing total persistence. The paper claims that this procedure identifies all minimum cluster sizes for which HDBSCAN* produces stable leaf clusters, that it is equivalent to 0-dimensional persistent homology on a novel metric, and that it achieves higher average ARI and lower sensitivity to the neighbour parameter k than HDBSCAN* on real-world datasets.

Significance. The paper has concrete strengths: a fully specified algorithm with pseudocode, an open-source implementation, a broad benchmark over 17 datasets, and a clear computational-cost comparison. The idea of reading all minimum-cluster-size changes from a single condensed tree is attractive and, if made correct, would be a useful contribution to parameter-light density-based clustering. However, the central equivalence claim is undermined by the leaf-tree construction rule in Sec. 4.1, and the sensitivity formula in Sec. 5.1 is degenerate as printed. These issues must be resolved before the empirical and conceptual claims can be accepted.

major comments (3)
  1. [Sec. 4.1, Eq. (5)-(6), Alg. C2] The leaf-lifetime rule is not faithful to HDBSCAN*. For two sibling clusters of sizes a=10 and b=20, Eq. (5) sets both children's s_max to min(a,b)=10 and Eq. (6) sets the parent's s_min to 10. At m_c=15, HDBSCAN* leaf selection keeps the larger child as a leaf: the smaller child is pruned as noise, and the parent has only one child, so the parent is not a leaf. The larger child should have s_max=20. This also contradicts Appendix A's d_s metric (Eq. 10): points in the two blobs are not in the same leaf until m_c>20, not m_c>10. Consequently the leaf tree, the persistence trace, and the selected m_c are built from incorrect lifetimes, and the claim that PLSCAN 'identifies all minimum cluster sizes for which HDBSCAN* produces stable (leaf) clusters' is unsupported. The s_max/s_min rules must be corrected (e.g., a child's s_max should be its own size and a parent's s_min the maximum child
  2. [Sec. 5.1, Eq. (9)] The sensitivity score in Eq. (9) is identically 1 because the numerator and denominator are the same expression: |ARI(k_i±Δk)-ARI(k_i)| / |ARI(k_i±Δk)-ARI(k_i)|. As printed, the formula cannot generate Fig. 6 or support the conclusion that PLSCAN is less sensitive to k. Please correct the denominator (likely ARI(k_i) or a related baseline) and rerun the sensitivity analysis.
  3. [Sec. 5.1, Experimental setup] ARI is computed only over non-noise points. Since PLSCAN and HDBSCAN* label different fractions of points as noise (the per-dataset noise-fraction curves in Appendix B make this visible), a method that labels many points as noise can obtain a high ARI on the remaining subset even if its full labeling is poor. The average ARI in Table 1 and the sensitivity distributions in Fig. 6 are therefore not directly comparable. Please report ARI on all points, or use a noise-aware metric, and provide noise fractions together with the ARI scores.
minor comments (3)
  1. [Appendix A] The metric d_s in Eq. (10) is defined directly in terms of the leaf clusters ℒ_s, so 0-dimensional persistent homology on (Y,d_s) recovers those leaf lifetimes by construction. This is a formalization, not an independent characterization; the text should state this more carefully and avoid implying that PH 'uncovers' the leaf lifetimes without already knowing them.
  2. [Sec. 4.1] The claim that merge sizes and distances do not change with the m_c threshold is asserted without proof. Distances along the original dendrogram are invariant, but leaf statuses are m_c-dependent; a precise statement with a proof is needed, especially in light of the major comment above.
  3. [Abstract / Sec. 1.2] The paper advertises a 'parameter-light' method, but PLSCAN still requires k, the initial minimum cluster size, the persistence-measure choice, and UMAP preprocessing hyperparameters. Please state the number and role of the remaining parameters precisely.

Circularity Check

1 steps flagged · score 4.0 of 10

The persistent-homology equivalence is definitional: the metric d_s encodes leaf lifetimes, so PH recovers them by construction; the leaf-tree algorithm and benchmarks are otherwise independent.

  1. self definitional [Appendix A.2-A.3 (Eqs. 10, 12, 14)]
    "we define a distance d_s(x_i,x_j) on X as the minimum pruning value s that places two points in the same leaf cluster: d_s(x_i,x_j)=min_s {s | ∃L∈ℒ_s : x_i,x_j∈L} ... ds(x_i,π_i)=min_s {s|∀L∈ℒ_s : x_i∉L} ... Applying persistent homology to the metric space (Y,d_s) uncovers the minimum cluster size ranges for which leaf-clusters exist."

    The metric is constructed directly from the leaf-cluster sets ℒ_s: d_s is, by definition, the first pruning level at which two points share a leaf, and π_i is the first level at which a point belongs to no leaf. The Vietoris-Rips zero-dimensional barcode therefore has births and deaths equal to the very s_min/s_max values that define ℒ_s. The claimed 'equivalence' reads back exactly the information put into the metric; it is a restatement, not an independent derivation of leaf lifetimes. Since Appendix A is presented as demonstrating that PLSCAN 'is equivalent to persistent homology,' this part of the derivation is circular with the definition of the metric.

full rationale

The central PLSCAN algorithm—constructing a leaf tree from a single condensed tree and selecting m_c by total persistence—does not fit a parameter to the evaluation labels, and the ARI/sensitivity comparisons use external ground-truth labels. The self-citations to McInnes et al. are standard implementations of HDBSCAN*, not load-bearing uniqueness claims. The only load-bearing 'unification' that reduces by construction is the persistent-homology framing in Appendix A. I also note that Eq. 5's s_max = min(child sizes) is a correctness concern for unbalanced merges relative to HDBSCAN*'s leaf selection, but that is a mismatch with HDBSCAN*, not a circular step, so it does not further raise the circularity score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The algorithm's core novelty is the leaf-tree/persistence-trace computation; it imports HDBSCAN*'s mutual-reachability machinery and adds a persistence-selection heuristic. The main free choices are k, initial m_c, persistence measure, and UMAP preprocessing. The persistent-homology equivalence is built into the metric definition (Appendix A), so it adds no independent evidence.

free parameters (4)
  • k (number of mutual-reachability neighbours) = 4 recommended default; 10 in runtime tests; range [2,50] in sensitivity analysis
    PLSCAN does not eliminate k; core distances and the mutual-reachability graph depend on it. The recommended k=4 is imported from DBSCAN/HDBSCAN* practice, not derived.
  • persistence measure choice = size (recommended)
    The paper evaluates five persistence measures and recommends 'size' because it achieved the highest average ARI on the benchmark; this is post-hoc selection on the test set, not a parameter-free prediction.
  • initial minimum cluster size m_c = k (default), e.g., 4
    Alg. 1 condenses the tree with m_c=k, and leaf tree s_min is initialized to k. Although PLSCAN auto-selects the final m_c, the initial condensation threshold is an input.
  • UMAP preprocessing hyperparameters = cosine distance, n_neighbors=min(250,N-1), repulsion strengths 0.001/0.01/0.1/1, project to min(50,D) dimensions
    The ARI benchmark is on UMAP projections; these choices affect clusterability and are fixed across datasets, but not justified as optimal or varied in sensitivity analysis.
assumptions (5)
  • domain assumption A single condensed tree built at low m_c contains all merges that occur at higher m_c, with unchanged merge sizes and distances.
    Sec. 4.1 and Fig. 2a. Central to computing the leaf tree from one tree. Believed true because pruning removes small branches, but not proven in the paper.
  • ad hoc to paper Leaf clusters form a monotonic hierarchy under increasing m_c, and the parent-birth rule in Eq. 6 (s_min = max of children's s_min) yields exact leaf lifetimes.
    Eqs. 5-6 define the leaf tree; the delayed parent birth is an algorithmic convention to produce nested segments, not a consequence of HDBSCAN*.
  • standard math The distance d_s in Eq. 10 is a metric on X and satisfies the ultrametric inequality Eq. 11; the extension to Y=X∪Π in Eqs. 12-14 is well-defined.
    Appendix A.2 asserts this without a proof; needed for the persistent-homology equivalence.
  • ad hoc to paper Total size persistence, summed over leaf clusters at a cut, is a valid proxy for clustering quality at that m_c.
    Sec. 4.2 defines this quality measure; it is a heuristic inspired by Leung et al. and Rieck & Leitte, not derived from an objective error.
  • domain assumption Monotonic pruning of a linkage hierarchy by minimum cluster size is a valid scale-space smoothing of the density profile.
    Sec. 2 and Fig. 1; analogy to scale-space clustering used to justify PLSCAN's interpretability.
invented entities (2)
  • Special elements π_i (Π = {π_i | x_i in X})
    purpose: Added to the metric space so that points that become noise are connected to π_i at their leaf-lifetime endpoint (Eqs. 12-14), making the filtration capture leaf disappearances.
    A bookkeeping device in Appendix A; no observable counterpart, and the filtration is defined to reproduce PLSCAN's lifetimes by construction.
  • Phantom root
    purpose: Represents the whole dataset as a root segment so the algorithm can handle precomputed minimum spanning forests and multi-component inputs (Alg. C2 lines 4, 19-21).
    An indexing device in the condensed/leaf tree; not a data cluster.

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Pith. "Pith review of Persistent Multiscale Density-based Clustering." pith.science (2026). https://pith.science/paper/2TNFU2J4

@misc{pith2026251216558,
  author       = {Pith},
  title        = {Pith review of: Persistent Multiscale Density-based Clustering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2TNFU2J4}},
  note         = {Machine review of arXiv:2512.16558}
}
abstract

Clustering is a cornerstone of modern data analysis. Detecting clusters in exploratory data analyses (EDA) requires algorithms that make few assumptions about the data. Density-based clustering algorithms are particularly well-suited for EDA because they describe high-density regions, assuming only that a density exists. Applying density-based clustering algorithms in practice, however, requires selecting appropriate hyperparameters, which is difficult without prior knowledge of the data distribution. For example, DBSCAN requires selecting a density threshold, and HDBSCAN* relies on a minimum cluster size parameter. In this work, we propose Persistent Leaves Spatial Clustering for Applications with Noise (PLSCAN), a multiscale density-based clustering algorithm that replaces HDBSCAN*'s fixed minimum cluster size pruning of a mutual-reachability linkage hierarchy with a persistence-based cluster selection procedure. Effectively, PLSCAN identifies all minimum cluster sizes for which HDBSCAN* produces stable (leaf) clusters. In concept, PLSCAN applies scale-space clustering principles and is equivalent to persistent homology on a novel metric space. We compare its performance to HDBSCAN* on several real-world datasets, demonstrating that it achieves a higher median ARI, is less sensitive to changes in the number of mutual reachability neighbours, and has higher stability under resampling. Additionally, we compare PLSCAN's computational costs to $k$-Means++, demonstrating competitive run-times on low-dimensional datasets. At higher dimensions, run times scale more similarly to HDBSCAN*.

Figures

Figures reproduced from arXiv: 2512.16558 by the authors.

Figure 1
Figure 1. OPTICS-like visualisation (Ankerst et al., 1999) demonstrating the minimum cluster size parameter’s smoothing effect. (a) A 2D point cloud from McInnes et al. (2022) with HDBSCAN* leaf-clusters for 𝑘 = 5 and 𝑚c = 100. (b) HDBSCAN*’s modelled density profile at multiple minimum cluster sizes. The 𝑥-axis contains all data points ordered to mimic the shape of probability density functions. The ordering is fixed for all… view at source ↗
Figure 2
Figure 2. Leaf tree construction explainer. (a) Condensed trees at multiple minimum cluster sizes for a 2D point cloud from McInnes et al. (2022). The tree at the lowest threshold contains all merges present in the later trees. The size and distance at which segments merge remain constant regardless of the threshold. (b) A simple condensed tree 𝑪 with two cluster merges and N = 150, annotated for the first cluster merge. Clus… view at source ↗
Figure 3
Figure 3. Novel PLSCAN concepts demonstrated on the data from Fig. 2a. (a) The leaf tree describes which local density maxima exist at each cluster size threshold. Colours indicate the top-10 highest total persistence peaks. Icicle widths encode the clusters’ excess of mass, i.e., the distance persistence sum over all points in the cluster (Campello et al., 2015). The clusters’ birth distances increase with the minimum cluste… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Clusterings for different values of 𝑘 on the data from [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: ARI–𝑘 curves for each algorithm configuration. Coloured lines indicate the algorithm configurations: HDBSCAN* with EOM in blue and leaf orange; PLSCAN with size in green, size-𝑑 in red, size-𝜆 in purple, 𝑑 in brown, and 𝜆 in pink. Lighter shaded lines indicate scores f…
Figure 6
Figure 6. Figure 6: Parameter sensitivity distributions over the evaluated datasets (Tab. 1). Sensitivity was computed using ARI scores following the procedure by Peng et al. (2022). Each violin shows 𝑘-perturbation sensitivities (Eq. 9) of a specific algorithm configuration, annotating t…
Figure 7
Figure 7. Figure 7: Quadratic regression lines relating computational cost to the number of data points and the number of dimensions. The left three plots show scaling over the number of data points in 2, 10, and 20 dimensions. The right-most plot shows scaling over the number of dimensio…

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.