Absolute continuity of generalized Wasserstein barycenters of finitely many measures

Source: arXiv:2605.06518 · Published 2026-05-07 · By Jianyu Ma

TL;DR

This paper addresses the absolute continuity of generalized Wasserstein barycenters on complete Riemannian manifolds when the transport cost takes the form c(x,y) = h(d_g(x,y)) for a strictly convex profile h that may lack C²-smoothness at the origin. The classical quadratic case (h(d) = d²/2) is well-understood: a single absolutely continuous marginal suffices to guarantee an absolutely continuous barycenter. For non-smooth costs such as h(d) = d^p/p with 1 < p < 2, the singularity of the Hessian along the diagonal creates an obstruction that, in the Euclidean literature (Brizzi–Friesecke–Ried, refs [4,5]), was handled by requiring all marginals to be absolutely continuous but relied critically on flat translational symmetry and coordinate-based implicit differentiation that does not transfer to curved manifolds.

The core contribution is a geometrically transparent three-step framework that (1) reduces the problem to a discrete approximation, (2) establishes local Lipschitz control of the inverse transport map F: z → x₁ on slices obtained by freezing (x₂,...,x_n) at discrete anchor points, and (3) handles the singular diagonal set by decomposing the support of the multi-marginal plan. This framework exposes precisely which analytic condition on h governs the assumptions: if h is C²-smooth up to the origin (h''(0) well-defined, e.g., p ≥ 2), a single absolutely continuous marginal µ₁ suffices; if h lacks this origin smoothness (e.g., 1 < p < 2), then all remaining marginals µ₂,...,µ_n must be absolutely continuous.

The result is a unified Theorem covering both cases on any complete Riemannian manifold (not necessarily compact), extending prior work of the same author (arXiv:2310.13832) from the quadratic case and extending the Euclidean results of [4,5] to the manifold setting. The proof strategy is constructive and explicit, relying on Jacobi field estimates, cut-locus avoidance, and a measurable selection argument, without invoking semi-concavity of the potential (which fails for p < 2).

Key findings

• For cost h(d) = d^p/p with p ≥ 2 (C²-smooth at origin, h''(0) finite), absolute continuity of a single marginal µ₁ is sufficient to guarantee an absolutely continuous generalized Wasserstein barycenter on any complete Riemannian manifold.
• For cost h(d) = d^p/p with 1 < p < 2 (Hessian scales as |x-y|^(p-2) → ∞ near diagonal, violating semi-concavity), absolute continuity of ALL marginals µ₁,...,µ_n is required; the paper proves this is the correct analytic boundary condition on h.
• The Euclidean proofs of Brizzi–Friesecke–Ried [4,5] implicitly rely on flat translational symmetry (specifically [5, (5.5) in Remark 5.1]) for inverse-Lipschitz bounds on the barycenter map; this reliance is identified as the obstruction to naive manifold generalization due to twisting of Jacobi fields.
• c-concave potentials associated with p-Wasserstein costs for p < 2 are proved NOT to be semi-concave (Remark 3.5), since the Hessian of h(d) = d^p/p scales as |x-y|^(p-2) → +∞ as x → y, contradicting the bounded-above Hessian requirement of semi-concavity.
• Cut-locus avoidance (Lemma 4.1, generalizing [11, Lemma 3.1]) is established for general strictly convex h satisfying (H1)-(H3): any h-barycenter z of a finitely supported measure avoids Cut(x_i) for all i, ensuring smoothness of y ↦ h(d_g(y,x_i)) near z.
• Injectivity of barycenter configurations (Lemma 4.2, generalizing [11, Lemma 3.5]): if two configurations x and x̃ in the support of a multi-marginal optimal plan share a common barycenter y, then x = x̃; this is proved via c-cyclical monotonicity and the strict convexity of h.
• The approximation framework (Section 5.1) reduces absolute continuity of the barycenter measure to the discrete case by approximating µ_i (i ≥ 2) with discrete measures, disintegrating γ into conditional slices, and passing to the limit — avoiding any global coordinate argument.

Methodology — deep read

Threat model and assumptions: This is a pure mathematics paper; there is no adversarial threat model. The assumptions are analytic: (M,g) is a complete Riemannian manifold (possibly non-compact, without boundary); the cost profile h: [0,∞) → [0,∞) satisfies (H1) h(0)=0, (H2) h is C² on (0,∞) with lim_{t↓0} h(t)/t = 0, (H3) h strictly increasing and strictly convex. The measures µ₁,...,µ_n have finite h-moments. The key dichotomy in assumptions is whether h extends to a C²-function at 0 (h''(0) := lim_{t↓0} h'(t)/t finite) or not.

Data and mathematical objects: There are no datasets. The objects are probability measures on M, multi-marginal optimal transport plans γ ∈ Π(µ₁,...,µ_n), c-concave potential functions ψ ∈ I_c(X,Y), and the generalized barycenter cost C(x₁,...,x_n) = inf_{y∈M} Σ λ_i h(d_g(y,x_i)). Compactly supported measures are used throughout the regularity arguments; the general case follows by localization.

Architecture / algorithm — Step 1 (Section 3, Optimal transport for general costs): The paper first extends McCann's theorem [16, Theorem 13] to the manifold setting for general h. The key tools are: (a) super-differentiability of x ↦ h(d_g(x,y)) at x ≠ y via the first variation of arc length (Lemma 3.3); (b) Lipschitz continuity of c-concave functions on compact sets (Lemma 3.4, using uniform Lipschitz constant L = h'(R) where R bounds all pairwise distances); (c) the Tangency Lemma (Lemma 3.6), which shows that if ψ is differentiable at x and (x,y) is in the c-superdifferential ∂^c ψ, then ∇ψ(x) = -h'(d_g(x,y))σ̇(0) where σ is the minimal geodesic from x to y — uniquely determining the transport map T(x) = exp_x(-(h')^{-1}(‖∇ψ(x)‖) · ∇ψ(x)/‖∇ψ(x)‖). Existence and uniqueness follow from Kantorovich duality and the fact that µ being absolutely continuous implies it charges no set of zero volume, hence the differentiability set of ψ has full µ-measure.

Architecture / algorithm — Step 2 (Section 4, Multi-marginal structure): Two structural lemmas are proved. Cut-locus avoidance (Lemma 4.1): for any configuration x = (x₁,...,x_n) ∈ supp(γ) and its h-barycenter z, z ∉ Cut(x_i) for all i. This uses a contradiction argument: if z ∈ Cut(x_i), the squared distance d²(·,x_i) fails to be semi-convex at z, and the Hessian of h(d(·,x_i)) inherits this singularity, violating the second-order optimality condition. Since d_g(z,x_i) > 0 for z ∈ Cut(x_i), the possible singularity of h at 0 is irrelevant. Injectivity (Lemma 4.2): uses c-cyclical monotonicity of supp(γ) to show that a shared barycenter y forces x₁ = x̃₁ via the identity λ₁(∇_y[h(d_g(y,x₁))] - ∇_y[h(d_g(y,x̃₁))]) = 0, and strict convexity of h ensures the gradient map x ↦ ∇_y h(d_g(y,x)) is injective.

Architecture / algorithm — Step 3 (Section 5, Absolute continuity — the core): The proof has three substeps. (5.1) Approximation: Given µ₁ absolutely continuous, approximate µ_i (2 ≤ i ≤ n) by discrete measures µ_i^k = Σ_j α_{ij} δ_{a_{ij}}. For each discrete approximation, the multi-marginal plan γ^k disintegrates as γ^k = Σ_j (weight) · γ^k_{(j)} where each slice γ^k_{(j)} has µ₁ as first marginal and fixed atoms (a_{2j},...,a_{nj}) as remaining marginals. The barycenter of the slice is B#γ^k_{(j)}, and the full barycenter is the weighted sum. The key reduction: it suffices to prove absolute continuity for each slice, which has n-1 marginals fixed at single points — reducing to a 2-marginal transport problem between µ₁ and a Dirac mass. (5.2) Lipschitz control: On the regular region (away from collisions z = x_i), one needs to bound the Jacobian of the inverse map F: z ↦ x₁. The paper establishes this via Jacobi field estimates on Riemannian manifolds. Specifically, the differential DF(z) is related to the Hessian of the barycenter cost C, and one shows ‖DF(z)‖ ≤ L(z) with a locally bounded L. In the smooth case (h''(0) finite), the Hessian of h(d_g(z,x_i)) extends continuously as z → x_i, so Lipschitz control holds globally. In the singular case (1 < p < 2), one must excise a neighborhood of the collision set {z = x_i}. (5.3) Decomposition: The support of γ is decomposed into a regular part (where the Jacobian bound holds) and a singular part (collision regions). For the singular part, the requirement that µ₂,...,µ_n be absolutely continuous ensures the collision sets {x_i = z} carry zero marginal mass, hence zero γ-mass. The Jacobian change-of-variables formula then gives absolute continuity of the pushed-forward barycenter measure.

Concrete end-to-end example: Consider n=2, M = S² (2-sphere), h(d) = d^(3/2)/(3/2) (p=3/2, so 1 < p < 2), µ₁ absolutely continuous, µ₂ absolutely continuous. The multi-marginal cost is C(x₁,x₂) = inf_{y} [λ₁ d(y,x₁)^(3/2)/(3/2) + λ₂ d(y,x₂)^(3/2)/(3/2)]. The barycenter map B: (x₁,x₂) ↦ z is well-defined on supp(γ) by Lemma 4.2. The inverse map F: z ↦ x₁ has a singularity wherever z = x₁ (collision). Since h''(t) = (3/2)(1/2)t^{-1/2} → ∞ as t → 0, the Hessian of h(d(·,x₁)) at z = x₁ blows up. By requiring µ₂ absolutely continuous, the set {(x₁,x₂): x₁ = B(x₁,x₂)} has γ-measure zero (since if x₁ = z then x₂ is determined and the event has measure zero under the product). On the complement, F is locally Lipschitz with bounded Jacobian, so B#γ ≪ Vol.

Reproducibility: This is a theoretical mathematics paper. No code, no weights, no datasets. All proofs are self-contained. The paper explicitly provides a detailed proof of McCann's theorem (Section 3) 'for self-completeness'. The approximation framework builds on the same author's prior work [14] (arXiv:2310.13832), which is publicly available.

Technical innovations

• A geometry-intrinsic three-step proof framework (approximate → Lipschitz control on slices → decompose support) that eliminates the flat-space implicit differentiation of Brizzi–Friesecke–Ried [4,5] and extends their Euclidean absolute continuity results to complete Riemannian manifolds without compactness.
• Precise identification of the analytic condition on h governing the marginal assumptions: the dichotomy between h''(0) finite (C²-smooth at origin, single absolutely continuous marginal suffices) versus h''(0) infinite (all marginals must be absolutely continuous), formalized in the main theorem's two cases.
• Extension of the Tangency Lemma and optimal transport map existence/uniqueness (McCann [16, Theorem 13]) to general strictly convex costs h(d_g) on complete non-compact Riemannian manifolds, without requiring semi-concavity of the potential (which fails for p < 2).
• A Jacobi-field-based inverse-Lipschitz bound for the barycenter map on Riemannian manifolds that replaces the coordinate Hessian calculations of [5, Remark 5.1] and is compatible with the curvature-induced twisting of Jacobi fields.
• Proof that c-concave potentials for p-Wasserstein costs with 1 < p < 2 are NOT semi-concave on any Riemannian manifold (Remark 3.5), closing a gap in the regularity theory of non-quadratic transport.

Limitations

• The paper establishes existence of absolutely continuous barycenters but does not address uniqueness of barycenters for general h (the uniqueness of the optimal transport plan is proved, but uniqueness of the minimizer of the barycenter functional in the non-quadratic case is not fully addressed in the truncated text).
• The Lipschitz control in Section 5.2 is established locally on 'regular regions' but the precise quantitative bounds on the Jacobian of F (and their dependence on curvature bounds of M) are not made explicit in the available text, making it unclear how the estimates degrade on manifolds of unbounded curvature.
• The framework requires compactly supported measures for the regularity arguments (Section 3 and 4), with the general finite-moment case handled by localization — the details of this localization step are not fully spelled out in the truncated text.
• No discussion of stability or continuity of the barycenter map with respect to perturbations of the marginals µ_i, which would be relevant for numerical approximation schemes.
• The assumption h'(0) = lim_{t↓0} h(t)/t = 0 (H2) excludes costs like h(d) = d (1-Wasserstein), so the framework does not cover the full range of practically interesting transport costs.
• The paper is purely theoretical with no numerical experiments or algorithmic implementation; computational complexity of computing these barycenters on manifolds under non-quadratic costs is not discussed.

Open questions / follow-ons

• Can the requirement that all of µ₂,...,µ_n be absolutely continuous (in the singular case 1 < p < 2) be weakened to, e.g., requiring only that the marginals are non-atomic or have no common atoms, as in some Euclidean results for related problems?
• Does the framework extend to infinite-dimensional spaces or metric measure spaces (e.g., Alexandrov spaces with curvature bounds) where the exponential map and Jacobi field theory have analogues but are less regular?
• What are the quantitative rates of convergence of the discrete approximation scheme (Section 5.1) in terms of the number of support points k, and can these be made sharp in terms of the geometry of M?
• Can uniqueness of the generalized Wasserstein barycenter (not just of the optimal transport plan) be established for general strictly convex h on Riemannian manifolds, possibly under additional assumptions on M such as non-positive sectional curvature?

Why it matters for bot defense

This paper is pure mathematical analysis with no direct application to bot defense, CAPTCHA, or machine learning systems. A bot-defense or CAPTCHA engineer would not find operational or implementation-level content here. The results concern the measure-theoretic regularity of a statistical aggregation operator (Wasserstein barycenter) under non-Euclidean geometry and non-quadratic transport costs, which is several abstraction layers removed from practical bot-detection.

That said, there is a loose indirect connection: Wasserstein barycenters are used in certain ML pipeline components that appear in behavioral biometrics and distributional robustness — for instance, computing a 'mean behavior profile' across heterogeneous user populations represented as probability measures, or domain adaptation under distribution shift. If such components use p-Wasserstein distances with p ≠ 2 (e.g., p = 1 for robustness to outliers), the theoretical guarantees proved here (absolute continuity of the aggregated measure, well-posedness of the barycenter) would underpin soundness of those operations on non-Euclidean feature spaces such as the manifold of covariance matrices or pose graphs. For a research engineer working at the intersection of geometric ML and bot detection, this paper provides foundational assurance that generalized Wasserstein aggregation is well-defined and regular under broader cost functions than the classical quadratic — but it offers no algorithms, no code, and no empirical validation.

Cite

@article{arxiv2605_06518,
  title={ Absolute continuity of generalized Wasserstein barycenters of finitely many measures },
  author={ Jianyu Ma },
  journal={arXiv preprint arXiv:2605.06518},
  year={ 2026 },
  url={https://arxiv.org/abs/2605.06518}
}

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• arXiv:2605.06518 (abstract)
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