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Clustering in hyperbolic balls

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper establishes a mathematical foundation for k-means and EM clustering inside hyperbolic balls, using a Möbius-invariant barycenter and Möbius mixture distributions.

desk verdict A useful clustering proposal for Poincaré balls, but the k-means update doesn't optimize its stated objective and the EM stationarity equation has a sign error; still worth a referee's time because the flaws are identifiable and likely fixable. read the letter →

arxiv 2501.19247 v1 pith:VK5O6WK7 submitted 2025-01-31 cs.LG

Vladimir Jaćimović , Aladin Crnkić This is my paper · ORCID
classification cs.LG MSC 62H3051M10
keywords k-meansexpectation-maximizationMöbiustransformationhyperbolicmachinelearningclusteringPoincaréballbarycentermixturemodel
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

This paper proposes that clustering in hyperbolic space can be put on the same footing as Euclidean clustering by replacing the arithmetic mean with a conformal barycenter and the Gaussian family with Möbius distributions. It develops two standard unsupervised-learning algorithms, k-means and expectation-maximization, for data living in the Poincaré ball $\mathbb{B}^n$ of any dimension. If correct, this gives practitioners principled clustering tools for embedded hierarchical data, because the barycenter and the Möbius family respect the geometry's symmetries. The paper also reports experiments on synthetic mixtures in the disc and in the three-dimensional ball, where the algorithms recover parameters in well-separated cases and fail on overlapping ones. The load-bearing assertion is that the weighted gradient-flow routine computes true weighted barycenters; the paper states this for weighted data without proof.

What carries the argument

The central object is the conformal barycenter, defined as the unique minimizer of the geodesically convex energy $\widetilde H_n(a) = -\sum_i w_i \log\big((1-|a|^2)(1-|y_i|^2)/\rho(y_i,a)\big)$, computed by the gradient-flow system (9)-(10) that transports the weighted points to a configuration whose weighted Euclidean sum is zero and then maps the origin back through the inverse Möbius transformation. The second pillar is the Möbius family of distributions with densities $p(x;a,s)\propto\big((1-|x|^2)(1-|a|^2)/\rho(a,x)\big)^s$, which is conformally invariant and transitively acted upon by the group of Möbius isometries of the Poincaré ball.

What would settle it

Take three points in the Poincaré disc with strongly unequal weights (for example, weights 100, 1, 1), compute the weighted barycenter by the paper's gradient-flow routine, and compare it with the numerical minimizer of (11) found by direct optimization; if they disagree, the center update in both algorithms can compute the wrong centers.

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Extended reading notes

Core claim

The paper's central claim is that a hyperbolic analogue of k-means and of Gaussian-mixture EM exists and is computable. For k-means the center of each cluster is the conformal barycenter, defined as the unique minimizer of a geodesically convex function; for mixtures the cluster model is the Möbius family of probability densities, and maximum likelihood estimates the location parameter as the weighted conformal barycenter. Both algorithmic steps — moving points by a hyperbolic gradient flow until their weighted sum vanishes and then pulling the origin back by the resulting Möbius transformation — are inherited from this geometry, so the cluster centers transform equivariantly under isometries of the ball. The paper further presents the EM equations, including a closed-form update for the concentration parameter in the disc and a digamma-function equation in higher dimensions, and reports experiments in $\mathbb{B}^2$ and $\mathbb{B}^3$.

Load-bearing premise

The weakest load-bearing step is the weighted barycenter computation: the paper assumes that running the gradient-flow system (9) with weighted points drives the weighted Euclidean sum to zero and that the Möbius preimage of zero is exactly the unique minimizer of the weighted objective (11), but it gives no proof or citation for the weighted case.

Editorial extensions

If this is right

  • Data embedded in a Poincaré ball of any dimension can be clustered by k-means with cluster centers updated by the weighted gradient-flow routine instead of the Euclidean average.
  • Mixtures of Möbius distributions can be learned by EM, with responsibilities computed from the Möbius density and parameters updated by weighted-barycenter and concentration-parameter estimation.
  • Because both the barycenter and the Möbius family are conformally invariant, the clustering outcome is equivariant under Möbius transformations: applying an isometry of the ball to the data applies the same isometry to the inferred cluster centers.
  • The reported experiments in the disc and the three-dimensional ball show that the algorithms recover mixture parameters close to ground truth when clusters are well separated; the same experiments show overlapping components are not recovered, so practical use requires diagnostics such as monitoring responsibilities and log-likelihood.
  • The M-step is numerically tractable: the concentration parameter has a closed-form update in the Poincaré disc and a digamma-equation solution in higher dimensions.

Reading between the lines

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

  • Beyond the paper, the same conformal-barycenter machinery could define other Euclidean algorithms on hyperbolic balls, such as k-medians or principal geodesic analysis, provided the corresponding objectives are geodesically convex; the paper does not address these.
  • The unproved weighted-convergence step suggests a testable mathematical conjecture: for any positive weights and initial points in the ball, the flow (9) with $f=(K/N)\sum_j \omega_j x_j$ drives the weighted sum to zero, and the limiting Möbius transformation pulls the origin to the weighted barycenter.
  • The reported failures on overlapping mixtures imply that, as with Euclidean EM, initialization and identifiability matter; a natural extension is an initialization strategy based on the unweighted barycenter or on k-means++ in hyperbolic distance, but the paper does not propose one.
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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

4 major / 4 minor

Summary. The paper proposes a mathematical framework for unsupervised learning in Poincaré balls. It defines conformal barycenters and weighted barycenters, introduces a family of Möbius distributions, and then presents two algorithms: k-means clustering whose centers are conformal barycenters, and an EM algorithm for mixtures of Möbius distributions. Experiments are reported for two- and three-dimensional balls. The central claim is that these algorithms provide rigorous, well-founded clustering tools for hyperbolic representations.

Significance. If correct, the paper would address a genuine gap: hyperbolic machine learning lacks standard unsupervised tools, and the conformal-invariance properties of the proposed barycenter and distribution family are appealing. The exposition is clear, and the experiments demonstrate that the proposed updates terminate and recover parameters in some simple settings. However, the paper relies on two unreviewed preprints by the same first author for the two central mathematical objects, and, more importantly, the algorithms as printed do not optimize the objectives they state. The k-means update minimizes a log-ratio barycenter objective rather than squared hyperbolic distance, and the n-dimensional EM M-step for concentration parameters has a sign error in the printed stationarity equation. These are load-bearing inconsistencies, not presentation issues.

major comments (4)
  1. [Section 5.1, Eq. (8), and Section 6.1] The k-means objective is stated as J = sum_i sum_j kappa_ij d_hyp^2(xi_i, a_j), but step iv replaces each center by the conformal barycenter, i.e. the minimizer of H_2 or H_tilde in Eq. (8)/(11). No theorem in the paper shows that the minimizer of (8) minimizes the sum of squared hyperbolic distances. In the one-dimensional real slice, stationarity for J gives artanh(a) = N^{-1} sum artanh(y_i), while stationarity for (8) gives N a/(1-a^2) = sum y_i/(1 - a y_i); these are different for generic configurations, so a Lloyd iteration can increase J. Section 6.1 inherits the same mismatch in all dimensions. The paper therefore does not implement the k-means objective it defines.
  2. [Section 6.2, Remark 3, Eq. (16)] The stationarity equation printed in Remark 3 has the digamma difference reversed. Differentiating (16) gives ψ(1+s-n/2) - ψ(1+s-n) = H_tilde(hat a) / (N sum_i w_i). The printed equation has the opposite order. Since ψ is increasing, for s > n-1 the printed left-hand side is negative while the right-hand side is nonnegative (and positive except in degenerate cases), so no solution exists. Thus the M-step for concentration parameters in the n-dimensional EM algorithm is inconsistent with the objective, and the B1/B2 results are not produced by the stated EM update.
  3. [Section 3.2, steps ii-iv] The weighted-barycenter routine is asserted without proof or citation. It is not shown that solutions of (9) with f = (K/N) sum_j w_j x_j drive sum_j w_j x_j(T) to zero, nor that the inverse Möbius preimage of zero minimizes the weighted objective (11). The unweighted case is attributed to [18], but the weighted case is used in every cluster-center update in both k-means and EM. The text also assumes strictly positive weights, while EM responsibilities can vanish, so the routine's range of applicability to the E-step is not stated.
  4. [Section 7] The statement that convergence 'can be established in an analogous way' to Euclidean counterparts is not supported. Euclidean Lloyd convergence relies on the centroid update minimizing the cluster objective, which is exactly the property missing here, and EM monotonicity relies on an exact M-step, which the current n-dimensional s-update does not provide. The paper cannot claim an analogous convergence argument without addressing the objective/update mismatch and the stationarity error.
minor comments (4)
  1. [Section 3.2, Definition 4] Definition 4 refers to the unique minimum of 'the function (8)', but the weighted objective should be (11).
  2. [Section 3.2] The weights are denoted w_i in the text and omega_j in the algorithm; the notation should be unified.
  3. [Section 6.2, iii-1] The mixing-probability update writes sum_{i=1}^n gamma_im; the index should be N, not n.
  4. [Sections 4 and 5.3] The central objects in Definition 3 and Eq. (13) are imported from the preprints [17] and [18] without restating or proving the required convexity, uniqueness, normalization, and MLE properties; the paper should either make these results self-contained or clearly mark them as imported assumptions.

Circularity Check

2 steps flagged · score 4.0 of 10

EM location updates are the barycenter objective by construction, and the barycenter apparatus is imported from load-bearing self-citations; the k-means objective mismatch is a correctness risk rather than circularity.

  1. uniqueness imported from authors [Section 3.1, pp. 3-4; ODE system (9)-(10) and steps iii-iv]
    "For a sufficiently large T, it holds that x1(T)+···+xN(T)=0. This means (see [18, Proposition 3]) that zero (center of the ball) is the conformal barycenter of points x1(T),...,xN(T). Furthermore, due to [18, Theorem 12], there exists a (unique up to an orthogonal transformation) Möbius transformation h ∈ Gn, such that x1(T)=h(y1),...,xN(T)=h(yN)."

    The computation of every conformal/weighted barycenter—hence every k-means cluster center and every EM location update—is justified by a uniqueness/computation theorem imported from [18], a preprint by the paper's own first author. The text supplies no proof of the weighted version (Section 3.2) and no independent check of [18, Prop. 3 / Thm. 12]. The algorithm's output is forced by that prior self-authored theorem, invoked as an external mathematical fact rather than derived or independently verified here.

  2. self definitional [Section 4, eq. (13); Section 4.2; Section 5.2 EM step ii-2]
    "p(x;a,s)=π^{n/2}Γ(1+s−n/2)/Γ(1+s−n)·((1−|x|²)(1−|a|²)/ρ(a,x))^s ... the maximum likelihood estimations â and ŝ are (see [18]): a) â is the conformal barycenter ... ii-2) Compute weighted barycenters a(j)m of points ξi ... with weights γim using the method explained in subsection 3.2."

    The Möbius density's dependence on the location parameter a is exactly exp(s·log((1−|a|²)(1−|x|²)/ρ(a,x))), so maximizing the likelihood over a is, by construction, minimizing H_n in eq. (8)—the same objective used to define the conformal barycenter in Definition 3. The EM M-step then 'computes weighted barycenters', i.e. repeats this same minimization with weights γim (eq. 11). Thus the EM location estimates are not an independent derivation from data; the barycenter definition is baked into the density, and the paper presents this identity as an MLE result imported from [18].

full rationale

Most of the paper's mathematical machinery is not derived inside the paper: the conformal barycenter (Definition 3), its gradient-flow computation, and the Möbius family (eq. 13) are taken from [17] and [18], both involving the first author. Those citations are load-bearing because every cluster-center update in k-means and every EM M-step calls the weighted-barycenter routine. I do not treat citation of prior mathematical work as circular per se, but here the cited preprints are neither machine-checked nor independently benchmarked, and the text does not reprove the key uniqueness and convergence facts. More sharply, the Möbius density (13) is defined by the same conformal factor that defines the barycenter objective (8), so the statement 'MLE â is the conformal barycenter' (Section 4.2) is true by construction; the EM's weighted-barycenter M-step then re-executes that same minimization. That makes the EM location learning partially definitional rather than an independent prediction. The k-means section, by contrast, is best described as an objective mismatch—the center update minimizes H_n while the stated objective J is squared hyperbolic distance—which is a correctness risk, not a circularity. The synthetic experiments on data generated from the same Möbius mixtures provide a self-contained sanity check but no external falsification. Overall, there is real self-citation load-bearing and one by-construction identity, but the algorithms also contain independent ingredients (assignment rules, responsibilities, concentration-parameter updates), so the circularity score is moderate rather than severe.

Assumptions & free parameters 4 free parameters · 6 assumptions · 2 invented entities

The framework is almost entirely built on the authors' prior papers [17] and [18]. The only genuinely new algorithmic content is the clustering wrappers: the k-means loop replaces Euclidean means by conformal barycenters, and EM uses the Möbius family's weighted MLE. The derivation introduces no new physical entities, but it inherits the unproven barycenter-flow and density-normalization assumptions from the prior work.

free parameters (4)
  • Initial concentration parameter s* = 2 (suggested default)
    Section 5.1 step i recommends s*=2 for sampling initial k-means centers from Moeb_2(0,s*); no justification or sensitivity analysis is given.
  • Concentration initializations s_i = sampled from Exp(1)+1
    Section 5.2 step i sets initial mixture concentrations randomly from Exp(1)+1; results depend on this random initialization.
  • Gradient-flow constant K = unspecified negative value
    The ODE system (9)-(10) requires a negative constant K; the paper never gives a value or rule, leaving the barycenter computation under-specified.
  • Number of clusters k = assumed known per experiment
    All experiments fix k equal to the true number of mixture components; the paper does not address model selection.
assumptions (6)
  • standard math H_n(a) in (8) is geodesically convex and has a unique minimizer
    Taken from [17]; not proved in this paper. This underpins the conformal barycenter definition and the k-means update.
  • standard math The ODE system (9)-(10) is the hyperbolic gradient flow for H_n and drives unweighted configurations to zero sum
    Taken from [18]; the paper relies on it for the barycenter algorithm without re-derivation.
  • ad hoc to paper The weighted version (system (9) with f = K/N Σ ω_j x_j) drives Σ ω_i x_i(T) to 0
    Asserted in Section 3.2 step iii without proof or citation; if false, all weighted barycenter updates in k-means and EM are wrong.
  • domain assumption The density in (13) is a valid probability density on Bn
    Normalization is claimed via the stated constant; the validity is taken from [18] and not verified in the text.
  • domain assumption Moeb_n(a,s) is conformally invariant and G_n acts transitively
    Used to justify sampling and MLE in Section 4; stated without proof, attributed to [18].
  • domain assumption The MLE equations (15)-(16) are concave with unique solutions
    The paper says 'it is easy to check' but provides no proof; the printed stationarity equation in Section 6.2 is inconsistent with (16), so this needs checking.
invented entities (2)
  • Conformal barycenter (Definition 3)
    purpose: Center of a point set in Bn used as the k-means cluster center
    Defined in [17], not this paper; it is a mathematical construction, and it is not shown to minimize the k-means squared-distance objective used in Section 5.1.
  • Möbius distributions Moeb_n(a,s)
    purpose: Probability families used as mixture components in EM
    Introduced in [18]; chosen so that the MLE of the location parameter is exactly the conformal barycenter, making EM convenient but providing no external falsifiable prediction.

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Cite this review

Pith. "Pith review of Clustering in hyperbolic balls." pith.science (2026). https://pith.science/paper/VK5O6WK7

@misc{pith2026250119247,
  author       = {Pith},
  title        = {Pith review of: Clustering in hyperbolic balls},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VK5O6WK7}},
  note         = {Machine review of arXiv:2501.19247}
}
abstract

The idea of representations of the data in negatively curved manifolds recently attracted a lot of attention and gave a rise to the new research direction named {\it hyperbolic machine learning} (ML). In order to unveil the full potential of this new paradigm, efficient techniques for data analysis and statistical modeling in hyperbolic spaces are necessary. In the present paper rigorous mathematical framework for clustering in hyperbolic spaces is established. First, we introduce the $k$-means clustering in hyperbolic balls, based on the novel definition of barycenter. Second, we present the expectation-maximization (EM) algorithm for learning mixtures of novel probability distributions in hyperbolic balls. In such a way we lay the foundation of unsupervised learning in hyperbolic spaces.

Figures

Figures reproduced from arXiv: 2501.19247 by the authors.

Figure 1
Figure 1. Barycenters of configurations of points in the Poin [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Weighted barycenters of configurations of points i [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Random samples in Poincaré disc: a) s = 3, a = 0; b) s = 5, a = 0; c) s = 7, a = 0; d) s = 3, a = 0.9e i π 4 ; e) s = 5, a = 0.9e i π 4 , and f) s = 7, a = 0.9e i π 4 . 4.1 Random variate generation For the sake of completeness we briefly expose the procedure for the random variate generation from the Möbius distributions proposed in [18]. Suppose that we want to sample a random point from Moeb(a ∗ , s∗ ), with give… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Experiment A1 (three panels): a) ground truth; b) k [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Log-likelihood for EM in Experiment A1 c) The third cluster: π3 = 1/4, a3 = −2i/3, s3 = 8; d) The forth cluster: π4 = 3/20, a4 = 1/2 + i/2, s4 = 10. Results K-means found the following barycenters: a1 = −0.0000426429 − 0.0070095i, a2 = 0.785931 − 0.0545432i, a3 = 0.003…
Figure 6
Figure 6. Figure 6: Experiment A2 (three panels): a) ground truth; b) k [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Log-likelihood for EM in Experiment A2 d) The forth cluster: π4 = 2/5, a4 = −0.5, s4 = 1.5. Results K-means found: a1 = 0.891161 − 0.0656324i, a2 = −0.8407 + 0.139705i, a3 = −0.209848 + 0.0666284i and a4 = −0.618712 − 0.456431i. EM found the following mixture of Möbius…
Figure 8
Figure 8. Figure 8: Experiment A3 (three panels): a) ground truth; b) k [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Log-likelihood for EM in Experiment A3 Results K-means found the following barycenters: a1 = −0.023886 − 0.0199269i, a2 = 0.247391 + 0.0562165i and a3 = 0.338258 − 0.334991i. EM did not recognize any clusters in this data sets. Instead, the algorithm suggested that all…
Figure 10
Figure 10. Figure 10: Experiment B1: a) ground truth; b) k-means; c) EM a [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Log-likelihood for EM in Experiment B1 s ∗ = 5 =⇒ 105 2  b 7 7 − 2 b 5 5 + b 3 3  = κ. The above formulae are used for generation of the data in this experiment, as well as in the next one. Results K-means found the following barycenters: a1 = {0.840602, −0.0144454,…
Figure 12
Figure 12. Figure 12: Experiment B2: a) ground truth; b) k-means; c) EM a [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: Log-likelihood for EM in Experiment B2 b) π2 = 0.25, a2 = {−0.4, −0.2, −0.3}, s2 = 5; c) π3 = 0.2, a3 = {0.6, −0.6, 0}, s3 = 4; d) π4 = 0.2, a4 = {0, 0, 0.85}, s4 = 3. Results K-means found the following barycenters: a1 = {−0.00723196, −0.445543, 0.401501}; a2 = {−0.3…

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Forward citations

Cited by 1 Pith paper

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