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Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces

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Pith's one-line read Classical Fréchet means and medians can flip under tiny perturbations, but the barycentric merge tree is Lipschitz-stable under Wasserstein perturbations and consistently estimable from samples.

desk verdict The barycentric merge tree is a real addition to non-Euclidean statistics and the main stability bound holds up, but the connectivity-constant hypothesis is under-advertised and Lemma 6.1 has a fixable constant error. read the letter →

arxiv 2505.09609 v2 pith:PTSZUTKW submitted 2025-05-14 math.ST math.MGstat.MEstat.TH

Washington Mio , Tom Needham This is my paper · ORCID
classification math.STmath.MGstat.MEstat.TH MSC 62R2062R3062R4055N31
keywords FréchetmeanbarycentermedianmodeestimationmergetreesWassersteindistancediffusionstatisticalconsistency
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

Fréchet means, medians, and modes on curved or general metric spaces are unstable statistics: on a circle, a tiny perturbation or a new sample can move the mean from one side to the other, and the instability does not vanish as the sample size grows. The paper replaces the single summary by a multiscale metric-measure object, the barycentric merge tree (BMT), whose leaves record connected components of local minima of the $p$-deviation function and whose internal nodes record how those components merge. Its central claim is a Lipschitz stability theorem: for connected, locally path-connected Polish spaces with an $L$-admissible pseudo-metric $\theta$ and a finite connectivity constant $K$, the functional Kantorovich–Sturm distance between two BMTs is at most $L(1+K) w_p(\mu,\mu')$. That inequality upgrades unstable point summaries to a provably stable structure and yields consistency plus explicit $n^{-1/s}$ convergence rates for BMTs built from empirical measures. Modes enter the same framework by taking $\theta$ to be a diffusion pseudo-metric derived from a kernel, so the stability guarantee covers mode merge trees as well.

What carries the argument

The object that carries the argument is the barycentric merge tree (BMT): the quotient space $\mathcal{T}_p(\mathcal{X})=X/{\sim}$, where $x\sim y$ when $x$ and $y$ lie in the same connected component of the sublevel set $\sigma_p^{-1}((-\infty,t])$ at level $t=\sigma_p(x)=\sigma_p(y)$. The quotient map $\alpha_p$ pushes $\mu$ to $\mu_p$, and the merge height defines the pseudo-metric $d_p$; the tree also carries the descending function $\kappa_p$. Three ingredients make stability work: Proposition 2.2 shows the $p$-deviation function is $\theta$-Lipschitz; Proposition 3.12 shows $\alpha_p$ is $KL$-Lipschitz whenever the space has connectivity constant $K$ and $\theta$ is $L$-admissible; and the Coupling Lemma (Lemma 4.4) converts a uniform bound $|\sigma_p-\sigma'_p|\le r$ into a metric coupling of the two trees with distortion at most $2r$. The final inequality follows by coupling the underlying measures with a Wasserstein-optimal transport plan and applying Minkowski's inequality to the structural and functional offsets.

What would settle it

On the Hawaiian earring, the closed union of circles tangent at the origin with radii $1/n$, the connectivity modulus is infinite: take $x_n$ and $y_n$ on adjacent circles at the leftmost points; their metric distance is $1/(n(n+1))$, while every path between them passes through the origin at distance about $1/n$ from both endpoints. Computing the BMT stability ratio on this space would show that no finite Lipschitz constant can hold without the $K$ assumption, so the theorem's scope is exactly the class of spaces with finite connectivity modulus.

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Extended reading notes

Core claim

The discovery is that the right object to estimate is not the set of minimizers of the deviation function but the whole merging pattern of its sublevel sets, made into a functional metric measure space. The barycentric merge tree $\mathcal{T}_p(\mathcal{X})$ is the quotient of $X$ by the relation that identifies points lying in the same connected component of a sublevel set $\sigma_p^{-1}((-\infty,t])$; the quotient map pushes $\mu$ forward to $\mu_p$, and the merge height defines a pseudo-metric $d_p$ with the property that the induced function $\kappa_p$ is 1-Lipschitz. Theorem 4.5 proves $D_{\mathrm{KS},p}(\mathcal{F}_p,\mathcal{F}'_p)\le L(1+K)w_p(\mu,\mu')$ for connected, locally path-connected Polish spaces, where $\mathcal{F}_p=(\mathcal{T}_p,d_p,\mu_p,\kappa_p)$. Because $K=1$ for geodesic spaces, the bound becomes $D_{\mathrm{KS},p}\le 2w_p$ when $\theta=d_X$. Corollary 4.7 then uses empirical Wasserstein convergence rates to conclude almost-sure consistency and $\mathbb{E}[D_{\mathrm{KS},p}]\le C\,\mathrm{diam}(X)\,n^{-1/s}$ for any $s$ above the upper $p$-Wasserstein dimension of $X$.

Load-bearing premise

The stability bound requires a finite connectivity constant $K$: every pair of points must be joinable by a path whose maximum distance to the two endpoints is at most $K$ times their distance, and connected Polish spaces need not satisfy this.

Editorial extensions

If this is right

  • On geodesic spaces with $\theta=d_X$, the stability bound is $D_{\mathrm{KS},p}\le 2w_p(\mu,\mu')$, so BMTs are Lipschitz-stable summaries rather than only continuous ones.
  • Empirical BMTs converge almost surely to the population BMT, with expected error at most $C\,n^{-1/s}$ for every $s$ above the upper Wasserstein dimension; sample size directly controls the accuracy of the topological summary.
  • Mode merge trees built from diffusion distances of uniformly Lipschitz kernels, including heat kernels on compact manifolds, inherit the same stability and consistency guarantees.
  • For compact spaces, a $\delta$-covering graph with edges between points at distance at most $3\delta$ yields a combinatorial BMT that is $C\delta$-close in the Kantorovich–Sturm distance, and a binning step keeps the simplified tree within $\varepsilon$.
  • The framework applies even when barycenters are highly non-unique, such as uniform antipodal masses on the circle, because the BMT records the full merging pattern instead of selecting one minimizer.

Reading between the lines

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

  • A natural two-sample test for distributions on manifolds would compare empirical BMTs in $D_{\mathrm{KS},p}$; the $n^{-1/s}$ rate gives a control on the null distribution, though the authors do not develop such a test.
  • The diffusion-kernel construction yields a one-parameter family of BMTs as the scale $t$ varies, and the paper notes phase transitions; choosing $t$ automatically from the largest gap in $D_{\mathrm{KS},p}$ across scales is a testable extension the authors leave open.
  • Because the bound scales linearly with the admissibility constant $L$, using a diffusion distance can inflate constants; whether a $\theta$-adapted Wasserstein distance would absorb $L$ and sharpen the rates is a natural next step.
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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 / 4 minor

Summary. The manuscript introduces the barycentric merge tree (BMT), a functional merge-tree summary of the p-deviation function of a probability measure on a metric space, and proposes it as a robust representation of barycenters and modes. The BMT is a quotient of the space by connected components of sublevel sets, equipped with a merge metric and pushforward measure; modes are treated as barycenters for diffusion pseudo-metrics. The main theoretical results are: a Lipschitz stability bound DKS,p(Fp,F'p) ≤ L(1+K) wp(μ,μ') (Theorem 4.5), empirical consistency with rates n^{-1/s} (Corollary 4.7), a discrete δ-approximation scheme with O(δ) error (Theorem 6.2 and Corollary 6.3), and a binning simplification step. Numerical examples on the circle, sphere, and polygon shape space illustrate the framework, and code is provided.

Significance. A stable, computable summary of barycenters and modes would be a meaningful advance, and the proof of Theorem 4.5 is clean and self-contained given standard optimal transport machinery, with explicit constants and a clear Coupling Lemma. The paper also gives a discrete approximation pipeline with publicly available code and derives explicit consistency rates from Weed–Bach; these are concrete strengths. However, the hypotheses needed for the main results are narrower than the abstract suggests: the finite connectivity constant K is not implied by the standing Polish-space assumptions, and the discrete approximation lemma contains a proof gap. The central stability claim itself appears sound, but the scope statements and the discrete approximation guarantee need correction.

major comments (3)
  1. [§2.2, Definition 2.3; §4, Theorem 4.5; §7] The standing hypotheses 'connected and locally path-connected Polish' do not imply the existence of a finite connectivity constant K. For the Hawaiian earring H = ⋃ C_n, where C_n is the circle of radius 1/n centered at (1/n,0), take x_n=(2/n,0) and y_n=(2/(n+1),0); then d_X(x_n,y_n)=2/(n(n+1)), but every path between x_n and y_n passes through (0,0), so r_X(x_n,y_n) ≥ 2/n and the ratio is at least n+1, giving K_H=∞. Since Proposition 3.12, Theorem 4.5, and Corollary 4.7 all require a finite K, the abstract's 'more general metric spaces' and Section 7's claim of results 'at the generality of all Borel probability measures on a Polish metric space' overstate the scope. The authors should add finite connectivity modulus to the standing assumptions in the abstract and Section 7, or prove the results under weaker assumptions.
  2. [§6.2, Lemma 6.1, Eq. (58)] The proof of the second inequality in Lemma 6.1 applies the connectivity condition to the wrong endpoints. Equation (58) bounds d_X(v,γ_i(t))∨d_X(γ_i(t),w), where v,w are the endpoints of the full path, but Definition 2.3 applied to the subpath from z_{i-1} to z_i would only justify a bound on d_X(z_{i-1},γ_i(t))∨d_X(γ_i(t),z_i). Consequently, the displayed conclusion 'd_X(γ(t), z_{i(t)}) ≤ ε+Kδ' does not follow from (58). Re-deriving the estimate with per-edge connectivity yields an additional factor, at least 3 since d_V(z_{i-1},z_i)≤3δ, so Lemma 6.1 and therefore Theorem 6.2 and Corollary 6.3 need a corrected proof or modified constants. The main stability theorem, Theorem 4.5, is unaffected.
  3. [§4, Corollary 4.7(ii)] The statement of Corollary 4.7(ii) omits the bounded-diameter hypothesis that its proof invokes. The proof cites Weed–Bach 'under the hypothesis that diam(X)≤1', but the corollary as stated asserts E[DKS,p] ≤ C diam(X) n^{-1/s} without any finiteness or boundedness condition on diam(X). If X is unbounded, the bound is not meaningful, and if diam(X) is only finite, the constant in the Weed–Bach estimate generally depends on the diameter. The bounded-diameter assumption (or an equivalent moment/support condition) should be added to the statement.
minor comments (4)
  1. [§4.2, heading] The heading of Section 4 refers to 'FMTs' while the rest of the paper uses 'BMTs'; the notation should be made consistent.
  2. [§3, Definition 3.5] The phrase 'cf. [13, 19]' after the definition is helpful, but the merge-height distance is essentially the cophenetic distance for merge trees; a brief explanatory sentence connecting these notions would improve readability.
  3. [§6.2, Theorem 6.2 proof] The proof of Theorem 6.2 refers to 'Proposition 3.12 and (6.2)' immediately before the structural-offset computation; the equation reference should be made explicit, since the displayed equation defining δ_r is not labeled in the text.
  4. [§7, last paragraph] The GitHub link is a useful reproducibility statement, but the corresponding repository should be cited in a stable archival form, such as a DOI or Zenodo record, for the published version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: BMT stability is a parameter-free derivation under explicit connectivity assumptions.

full rationale

The central derivation is self-contained. Theorem 4.5 is proved directly: equations (36)-(38) bound the pointwise deviation difference by L times the Wasserstein distance using Minkowski's inequality and L-admissibility; the Coupling Lemma (Lemma 4.4) converts that bound into a metric coupling on the merge trees; Proposition 3.12 uses the finite connectivity constant K to make the quotient map alpha_p KL-Lipschitz; and equations (39)-(42) assemble the final Lipschitz bound. No fitted parameter is relabeled as a prediction: L and K are explicit standing hypotheses, and the Wasserstein distance is an independent external quantity, not a function of the barycentric merge trees. Corollary 4.7 imports the empirical Wasserstein convergence estimate of Weed and Bach [38], which is external to the authors' prior work. Citations to [13] and [14] concern standard merge-tree metrics and Reeb-space arguments, but the key proof of Proposition 3.8 is reproduced rather than assumed, and no ‘uniqueness theorem' or unverified self-citation is load-bearing. Citation [15] supplies a routine diffusion-distance formula, but the admissibility of the heat-kernel diffusion distance is also directly established in Example 5.7. The finite connectivity constant K is a genuine substantive hypothesis: some connected, locally path-connected Polish spaces have infinite connectivity modulus, so the abstract's and Section 7's broad phrasing overstates the scope of Theorem 4.5 and Corollary 4.7. That is a scope/correctness caveat, not circularity, because the theorem states the assumption rather than hiding the conclusion in it. Similarly, the alleged proof issue around inequality (58) in Lemma 6.1 would be a constant-factor correctness matter, not a reduction of a conclusion to its input. Overall, the claimed stability and consistency results are derived, not defined, into existence.

Assumptions & free parameters 2 free parameters · 9 assumptions · 1 invented entities

The paper is largely self-contained: proofs are given for the key stability and approximation results. The framework relies on standard results in optimal transport, merge-tree/Reeb-space theory (some from the authors' own prior papers), and classical manifold facts. No parameters are fitted to make the theorems hold; the diffusion scale t and discretization parameters δ, ε are user-chosen modeling or accuracy controls, not fitted constants.

free parameters (2)
  • Diffusion scale t = user-selected; not fitted
    In Section 5, mode merge trees use diffusion distances θ_k,q from kernels k_t, with t>0 a scale parameter chosen by the user (heat kernel or exponential kernel). Stability holds for any t for which θ is admissible, so the theoretical results do not depend on a fitted value.
  • Grid spacing δ and binning level ε = user-selected accuracy controls
    In Section 6, the discrete approximation uses a δ-covering of the compact space and a binning threshold ε; the resulting error bounds are O(δ) and O(ε), and no particular values are needed for the central claim.
assumptions (9)
  • standard math Minkowski inequality and standard Lp norm properties
    Used in Proposition 2.2 and in the proof of Theorem 4.5 to pass from pointwise deviation bounds to Wasserstein bounds.
  • standard math Properties of Wasserstein distances and Kantorovich-Sturm distances (Sturm 2006, Mémoli 2011)
    Used to define DKS,p and to invoke metric-coupling arguments in Lemma 4.4 and the Stability Theorem.
  • standard math Burago-Burago-Ivanov proposition on metric couplings (Prop 4.3)
    Used to turn a correspondence with distortion at most 2r into a pseudo-metric δr on the disjoint union of the merge trees.
  • standard math Weed-Bach empirical Wasserstein convergence rate E[w_p(µ,µ_n)] ≤ C n^{-1/s}
    Used in Corollary 4.7 to obtain the explicit convergence rate for empirical BMTs.
  • domain assumption The space (X,d_X) is connected, locally path-connected and Polish, with finite connectivity modulus K
    Assumed in Theorem 4.5 and throughout; guarantees αp is Lipschitz and the stability bound is finite.
  • domain assumption The pseudo-metric θ is L-admissible (Lipschitz with respect to d_X)
    Required by Proposition 2.2 and Theorem 4.5; for mode applications, admissibility comes from Lipschitz kernels (Prop 5.5) or explicit heat-kernel bounds (Example 5.7).
  • domain assumption Heat kernel Lipschitz estimates on closed manifolds (Kasue-Kumura) and finiteness of reference measure ν
    Used in Example 5.6 and Proposition 5.5 to ensure diffusion distances from heat kernels are admissible on compact manifolds.
  • domain assumption Compactness and finite δ-covering for the discrete model
    Section 6 assumes X compact for the δ-covering construction and the error bounds of Theorem 6.2.
  • domain assumption For the shape-space example, the moduli space of n-gons is identified with Grassmannian Gr2(R^n) with its canonical metric
    Assumed in Example 5.3 to define the diffusion kernel and compute geodesic distances; stated via references [24,10].
invented entities (1)
  • Barycentric merge tree (BMT) independent evidence
    purpose: A measured metric tree representation of the sublevel-set merging structure of the p-deviation function, summarizing local and global barycenters or modes.
    Introduced in Definitions 3.1 and 3.11. It is supported by the stability theorem, consistency corollary, discrete approximation bounds, and the provided code repository; it is a formal construction, not an unverified postulate.

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Pith. "Pith review of Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces." pith.science (2026). https://pith.science/paper/PTSZUTKW

@misc{pith2026250509609,
  author       = {Pith},
  title        = {Pith review of: Robust Representation and Estimation of Barycenters and Modes of Probability Measures on Metric Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTSZUTKW}},
  note         = {Machine review of arXiv:2505.09609}
}
read the original abstract

This paper is concerned with the problem of defining and estimating statistics for distributions on spaces such as Riemannian manifolds and more general metric spaces. The challenge comes, in part, from the fact that statistics such as means and modes may be unstable: for example, a small perturbation to a distribution can lead to a large change in Fr\'echet means on spaces as simple as a circle. We address this issue by introducing a new merge tree representation of barycenters called the barycentric merge tree (BMT), which takes the form of a measured metric graph and summarizes features of the distribution in a multiscale manner. Modes are treated as special cases of barycenters through diffusion distances. In contrast to the properties of classical means and modes, we prove that BMTs are stable -- this is quantified as a Lipschitz estimate involving optimal transport metrics. This stability allows us to derive a consistency result for approximating BMTs from empirical measures, with explicit convergence rates. We also give a provably accurate method for discretely approximating the BMT construction and use this to provide numerical examples for distributions on spheres and shape spaces.

Figures

Figures reproduced from arXiv: 2505.09609 by the authors.

Figure 1
Figure 1. Barycentric merge tree example. Left: A probability density on the circle. The points marked by ⋆ are local minima of σ2, with the darkest indicating the (global) Fr´echet mean. Right: The barycentric merge tree summarizes the local minima, together with the shape of deviation function. In spite of the above, as is illustrated in Ex￾amples 1.1 and 1.2, the global minima of σp (or equivalently, Vp) can exhibit unstab… view at source ↗
Figure 2
Figure 2. Instability of global minima of σp on the circle. (a) A pdf on the circle, its corresponding deviation function σ2, and the global minimizers of σ2. The deviation function and the pdf are both rescaled in this plot for visual clarity. (b) Four draws of 100 samples from the distribution shown in (a); σ2 is plotted for each empirical distribution, along with its global minimizer, which varies wildly between samples. (… view at source ↗
Figure 3
Figure 3. Stability of Barycentric Merge Trees on the circle. (a) A circle with a bimodal distribution, its deviation function σ2 and the Fr´echet mean set, as in [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Fr´echet variance functions associated with the heat kernel at different scales for a pdf on the real line. As the notions of pockets of mass or clusters are inherently scale dependent, in mode detection, we employ diffusion distances θ derived from ker￾nel functions k…
Figure 5
Figure 5. Figure 5: Merge trees for modes: each pair shows the distribution on the circle from [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: BMTs for distributions on the sphere (see Example 3.2). Left: Heat map representation of the pdf of a highly symmetric distribution on the 2-sphere, with modes at each intersection of the sphere with the coordinate axes. The BMT for p = 2 is shown, together with points…
Figure 7
Figure 7. Figure 7: Quantitative stability of BMTs (see Example 4.6) Top Row: A summary of the exper￾iment. We begin with a smooth bimodal distribution on the circle, which is then sampled n times (n ∈ {10, 25, 50, 100, 250, 500, 1000}), from which we compute a Fr´echet mean (i.e., a poin…
Figure 8
Figure 8. Figure 8: Mode trees in shape space (see Example 5.3 for details). (a) Random samples from the moduli space of 10-sided polygons. We work with a dataset of 2000 samples. (b) A pdf is defined as the sum of kernels which decay exponentially in the geodesic distance from the polygo…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.