{"id":"7a5423a7-67b7-46be-93ee-0266db89b6bd","arxiv_id":"2505.09609","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Barycentric merge trees summarize Fréchet means and modes of distributions on metric spaces, and they are provably stable and consistently estimable under Wasserstein perturbations.","lead":"This paper introduces barycentric merge trees, a multiscale tree-shaped summary of where a distribution's means, medians, and modes sit on curved spaces. The authors prove that these trees change only a little when the distribution changes a little, and that empirical estimates converge at explicit rates.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Finite connectivity constant K is a real, non-automatic restriction: compact locally path-connected Polish spaces (e.g., the Hawaiian earring) have infinite K, so the abstract's unqualified 'metric spaces' scope exceeds Theorem 4.5.","rationale":"The reader's weakest assumption correctly identifies the finite connectivity constant K as the most load-bearing restriction on the central claim. My stress-test confirms this with an explicit compact counterexample: the Hawaiian earring satisfies all standing topological hypotheses (connected, locally path-connected, Polish) but has infinite connectivity modulus. Hence the main stability theorem does not cover all spaces that the abstract and concluding remarks suggest. The theorem as stated with the K hypothesis appears mathematically sound; the proof of Theorem 4.5 is coherent and the Lipschitz argument through Proposition 3.12 works under the stated assumptions. The additional gap in Lemma 6.1 is genuine — the displayed inequality (58) is not implied by the connectivity constant for the edge endpoints — but the error only weakens the constants in the discrete approximation bounds and does not threaten the qualitative approximation guarantee or the main stability estimate. Because the concern is a scope overstatement and a fixable proof error rather than a refutation of the central result, the existing CONDITIONAL verdict remains appropriate; I recommend no change.","tokens_in":24962,"tokens_out":21350,"duration_ms":220880,"concrete_test":"Compute the connectivity modulus of the Hawaiian earring H: for x_n = (2/n,0) and y_n = (2/(n+1),0), show that r_H(x_n,y_n) ≥ 2/n because any path joining the two circles must pass through the common point (0,0), while d_H(x_n,y_n) = 2/(n(n+1)); therefore r_H/d_H → ∞ and K_H = ∞. This confirms that Theorem 4.5 does not apply to a compact connected locally path-connected Polish metric space, so any unqualified statement about 'metric spaces' in the abstract or conclusions must be revised to require finite connectivity modulus. For completeness, re-derive inequality (58) with z_{i-1}, z_i in place of v, w to check that the Lemma 6.1 constant becomes 3KLδ rather than KLδ.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central stability estimate Theorem 4.5 and the consistency result Corollary 4.7 both require a finite connectivity constant K (Definition 2.3); this enters through Proposition 3.12, which makes the quotient map αp Lipschitz. The standing hypotheses 'connected and locally path-connected Polish' do not imply K < ∞. The Hawaiian earring H = ⋃_n C_n, where C_n is the circle of radius 1/n centered at (1/n,0), is compact, connected, locally path-connected and Polish. For x_n = (2/n,0) and y_n = (2/(n+1),0), the Euclidean distance is 2/(n(n+1)), but every path in H between them must pass through the common tangency point (0,0), so r_X(x_n,y_n) ≥ 2/n; the ratio is at least n+1, hence K_H = ∞. Thus Theorem 4.5 and Corollary 4.7 do not apply to such spaces, despite the abstract's 'Riemannian manifolds and more general metric spaces' and Section 7's claim to cover 'all Borel probability measures on a Polish metric space'. This is a scope overstatement, not an internal inconsistency in the theorem as stated. Separately, the proof of Lemma 6.1 contains an erroneous inequality (58) bounding dX(v,γ_i(t))∨dX(γ_i(t),w) instead of the relevant endpoint distances; re-deriving with the connectivity constant on each edge gives a factor of 3 (or more) in the constants of Theorem 6.2 and Corollary 6.3. The main stability theorem is unaffected by this second issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":25318,"tokens_out":6758,"duration_ms":66277,"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":[{"comment":"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.","section":"§2.2, Definition 2.3; §4, Theorem 4.5; §7"},{"comment":"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.","section":"§6.2, Lemma 6.1, Eq. (58)"},{"comment":"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.","section":"§4, Corollary 4.7(ii)"}],"minor_comments":[{"comment":"The heading of Section 4 refers to 'FMTs' while the rest of the paper uses 'BMTs'; the notation should be made consistent.","section":"§4.2, heading"},{"comment":"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.","section":"§3, Definition 3.5"},{"comment":"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.","section":"§6.2, Theorem 6.2 proof"},{"comment":"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.","section":"§7, last paragraph"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Main take: the barycentric merge tree is a real addition to the toolkit for non-Euclidean statistics, and the central stability bound (Theorem 4.5) is correct as far as I can see. The proof is clean and self-contained given standard optimal transport facts. The pushforward probability measure on the merge tree is a nice twist, and treating modes as barycenters with diffusion distances is a useful unifying move. The consistency corollary follows naturally from Weed–Bach rates. Credit also for shipping code.\n\nWhere I'd push back: the advertised scope is broader than the theorems. Stability needs a finite connectivity constant K (Definition 2.3). This is not automatic for connected locally path-connected Polish spaces — the Hawaiian earring is compact and has K = ∞. So the abstract's 'metric spaces' and Section 7's 'all Borel probability measures on a Polish metric space' overstate the domain. The authors should flag the hypothesis in the abstract and adjust the closing claims. This is a scope mismatch, not an error in the proof.\n\nSecond, Lemma 6.1 has a gap in the proof of the reverse inequality. In (58) they use the connectivity constant for a path from v to w, but the path at that point runs from z_{i-1} to z_i, so the bound is not justified as written. The fix I'd expect gives an extra factor in the constants of Theorem 6.2 and Corollary 6.3, likely 3 rather than 1. Main theorem unaffected. It's a small repair, but it needs to be made.\n\nThe citation pattern is fine: earlier merge-tree work by the authors and others is cited, and the claimed new results aren't in those references.\n\nNet: this is a solid paper worth refereeing. The stability theorem alone is a genuine contribution. The discrete approximation needs a corrected proof, and the connectivity hypothesis should be stated honestly. For a reader in statistical topology or shape analysis, it's useful. I'd send it to review with a request for revisions rather than desk-reject.","headline":"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.","tokens_in":25831,"tokens_out":2278,"would_cite":true,"duration_ms":22335,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R20","62R30","62R40","55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Fréchet mean","barycenter","median","mode estimation","merge trees","Wasserstein distance","diffusion distance","statistical consistency"],"falsifier":"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.","tokens_in":24755,"feed_emoji":"🌳","tokens_out":16621,"duration_ms":164750,"temperature":0.7,"pith_summary":"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.","feed_headline":"Barycentric merge trees make means and modes stable","feed_subtitle":"Replacing a single mean or mode by a multiscale merge tree gives Lipschitz stability and fast empirical convergence.","key_machinery":"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.","core_discovery":"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 $X$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the empirical Wasserstein convergence rates that Corollary 4.7 converts into consistency and $n^{-1/s}$ bounds.","marker":"[38]"},{"why":"Introduces the Kantorovich–Sturm $L_p$-transportation distance used to quantify BMT variation.","marker":"[35]"},{"why":"Supplies the distortion criterion for metric couplings used in the Coupling Lemma and discrete approximation proofs.","marker":"[9]"},{"why":"Defines the merge-tree metric structure and functional decoration that the BMT inherits from the deviation function.","marker":"[13]"},{"why":"Introduces $p$-deviation functions and their topological study, the starting point for the BMT construction.","marker":"[23]"},{"why":"Defines diffusion distances used to treat modes as barycenters within the same stability framework.","marker":"[12]"}],"fun_headline_variants":["Barycentric merge trees replace unstable means and modes","Stable barycenters: a multiscale tree representation","Means and modes get a robust merge tree structure","From fragile means to stable merge tree barycenters","Lipschitz stability for barycenters via merge trees"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Barycentric merge trees replace unstable means and modes","Stable barycenters: a multiscale tree representation","Means and modes get a robust merge tree structure","From fragile means to stable merge tree barycenters","Lipschitz stability for barycenters via merge trees"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00073,"raw_usage":{"total_tokens":3321,"prompt_tokens":1052,"completion_tokens":2269,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":668,"completion_tokens_details":{"reasoning_tokens":2189}},"tokens_in":668,"tokens_out":2269,"duration_ms":16980,"temperature":1.0,"reasoning_tokens":2189,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:27:40.830549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Weed and F","cited_arxiv_id":null,"evidence_quote":"Supplies the empirical Wasserstein convergence rates that Corollary 4.7 converts into consistency and $n^{-1/s}$ bounds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Kantorovich–Sturm $L_p$-transportation distance used to quantify BMT variation."},{"cited_title":"Burago, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the distortion criterion for metric couplings used in the Coupling Lemma and discrete approximation proofs."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces $p$-deviation functions and their topological study, the starting point for the BMT construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines diffusion distances used to treat modes as barycenters within the same stability framework."}],"review_version":1}