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arxiv: 2605.19305 · v1 · pith:TEWWUW6Enew · submitted 2026-05-19 · 💻 cs.GR · cs.CV· cs.LG

Mat\'ern Noise for Triangulation-Agnostic Flow Matching on Meshes

Tianshu Kuai , Arman Maesumi , Daniel Ritchie , Noam Aigerman This is my paper

Pith reviewed 2026-05-20 02:49 UTC · model grok-4.3

classification 💻 cs.GR cs.CVcs.LG
keywords flow matchingMatérn noisetriangulation agnosticismGaussian random fieldsmesh signal generationPoissonNetgenerative models on meshesspectral invariance
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The pith

A discretization of the Matérn Gaussian random field supplies a triangulation-agnostic noise distribution for flow matching on meshes.

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

The paper defines triangulation agnosticism of a distribution by the requirement that its spectrum remain unchanged under different triangulations of the same surface. It then shows that a suitable discretization of the Matérn process satisfies this spectral invariance and admits a simple sampling algorithm. The authors integrate this noise into a flow-matching pipeline that uses PoissonNet to learn signals directly in the gradient domain. Experiments demonstrate that the resulting models generate realistic outputs on meshes of more than one million triangles for tasks such as sampling elastic rest states and synthesizing humanoid poses.

Core claim

Triangulation agnosticism is formalized as spectral invariance of the noise distribution; a discretization of the Matérn Gaussian random field meets this definition and yields an efficient sampling procedure that, when used inside flow matching with a PoissonNet denoiser, produces generative models whose outputs are independent of the particular triangulation chosen for any given mesh.

What carries the argument

The discretization of the Matérn Gaussian random field, which preserves the continuous spectrum across arbitrary triangulations and supplies direct sampling.

If this is right

  • A model trained once can be applied to any triangulation of a given mesh without retraining.
  • Meshes containing more than one million triangles can be processed at practical cost.
  • Generation quality and diversity surpass prior mesh-based methods on elastic shapes and articulated poses.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spectral test for agnosticism could be used to certify other noise models on irregular domains such as point clouds or graphs.
  • Pre-training on one family of meshes and deployment on meshes with different connectivity becomes feasible without domain adaptation.
  • The same noise construction may extend directly to generative models for time-dependent or dynamic mesh signals.

Load-bearing premise

That invariance of the distribution's spectrum under changes in triangulation supplies the right and sufficient criterion for triangulation agnosticism.

What would settle it

If samples drawn from the Matérn discretization exhibit measurably different spectra or statistics when computed on two distinct triangulations of the identical underlying surface, the claimed agnosticism fails.

Figures

Figures reproduced from arXiv: 2605.19305 by Arman Maesumi, Daniel Ritchie, Noam Aigerman, Tianshu Kuai.

Figure 1
Figure 1. Figure 1: Generating different deformations of a high-resolution mesh through triangulation-agnostic flow matching. Top left: our method generates signals (visualized via colors) on meshes via the flow matching [Lipman et al. 2022] paradigm’s denoising process. Bottom left: in this case the signals generated correspond to a deformation of the mesh’s vertices. The model was trained on a dataset of physical simulation… view at source ↗
Figure 2
Figure 2. Figure 2: Matérn noise on different triangulations. Our noise exhibits similar structure across differently triangulated regions, as opposed to Naïve sampling from iid Gaussians for each vertex. our method attains state-of-the-art results, at a resolution that far exceeds prior generative methods for meshes. In designing a triangulation-agnostic generative framework, we ob￾serve that simply using an existing triangu… view at source ↗
Figure 3
Figure 3. Figure 3: Architecture of our triangulation-agnostic generative pipeline. We first perform noise sampling (bottom left) from our proposed triangulation-agnostic Matérn noise distribution via Algorithm 1. We then use that noise as the initial signal f0 in a flow matching denoising process (top), which iteratively denoises the signal into the final sample f1. In each flow step (bottom right) we input the current signa… view at source ↗
Figure 4
Figure 4. Figure 4: Empirical per-frequency statistics of Matérn noise. For each of three spectral coefficients – bf2,bf50,bf500 – we compute its distribution over 100k sampled noises, visualized as a histogram. We show overlays of the distributions for three mesh resolutions. Our method generates near￾identical distributions for all meshes, with decreasing variance for higher coefficients. Naïve, iid sampling of Gaussians, y… view at source ↗
Figure 5
Figure 5. Figure 5: Spectra of the differ￾ent noise choices. In geometry processing terms, 𝑓 is sam￾pled by first drawing a white noise sam￾pleW (which has iid Gaussian spectrum), and then solving a screened Poisson equa￾tion with respect to it, which acts as a low-pass filter for the spectrum. For the sake of our application, the screening term 𝜏 serves to introduce higher frequency content as desired (see Figures 5 and 2): … view at source ↗
Figure 6
Figure 6. Figure 6: Generated reposings. Our generative model, trained only on the 18k mesh, reposes humans to plausible poses solely via its learned representation. ALGORITHM 2: Normalized Matérn Noise Input :Mass matrix M, Laplacian L, user-chosen parameter 𝑐. 1 Compute the normalization factor Γ = | |M−1L| |𝐹 . 2 Compute the normalized screening term 𝜏 = Γ𝑐. 3 Compute the signal f via Algorithm 1, using 𝜏. 4 Return the nor… view at source ↗
Figure 7
Figure 7. Figure 7: Fixed 𝜏 vs. normalized 𝜏. The Matérn noise sampled using a fixed screening term 𝜏 is not invariant to the scaling of the mesh (global scaling factors of 0.1 and 2.0); with our proposed normalization scheme, the noise exhibits a consistent, scale-independent spectrum. linearly proportional to the eigenvalues. Thus, the user chooses the screening term hyperparameter 𝑐 ∈ R + , and we define 𝜏 = 𝑐Γ, yielding a… view at source ↗
Figure 8
Figure 8. Figure 8: Generalization to various 3D shapes. Our trained model generalizes to various humanoid meshes (never seen during training) [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Closest training sample and regressed SMPL meshes. Our gen￾erated samples approximate the GT deformation distribution well, and can be matched almost perfectly with the SMPL model, via regression. The closest training sample is quite far, showing our method generalizes. where M and 𝑨 are the diagonal matrices of vertex masses and face areas, respectively, 𝜹 = f1 −f0 is the GT velocity, and ∇ is the spatial… view at source ↗
Figure 10
Figure 10. Figure 10: Elastic equilibrium states of a deflated bunny. Our method generates diverse samples on the 70k resolution, though trained solely on 9k resolution [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Timings for generating elastic rest states over varying resolutions. Our method produces comparable results on all resolutions, even though it was trained solely on the 10k resolution. Generation time reported underneath each resolution [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Closest sample in the training set to each generated result in the elastic deformation experiment [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Elastic equilibrium of a fish with thin features. Our method produces plausible deformations even in the thin regions of the fins. model by simple gradient descent on the SMPL parameters w.r.t. vertex distance to the generated sample. We visualize the regressed SMPL model in [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparisons and ablations. We use the reposing experiment (Section 5.1.1) to compare to other mesh-based generative methods: MDF [Elhag et al. 2024], DoubleDiffusion [Wang et al. 2025a], the explicit method to compute Matérn noise, and ablations on other types of noise and screening terms [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparisons to MDF [Elhag et al. 2024] and DoubleDiffu￾sion [Wang et al. 2025a] on GMM. As the competing methods were trained on a regular mesh, MDF and DoubleDiffusion fail to predict correct signals on the irregular test mesh, as shown by the average error between the GT distribution and the generated samples. angular speed on a ground plane, until it reaches equilibrium. We then extract the boundary tr… view at source ↗
Figure 16
Figure 16. Figure 16: Generated results on topologically corrupted source meshes. Trained solely on the uncorrupted source mesh, our method still produces valid results when parts of the source mesh are removed, exhibiting artifacts only near the modified boundaries [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Results on source meshes of deteriorated triangulation qual￾ity. We perform 1k and 3k edge flips to create badly triangulated source meshes to test our model, trained solely on uncorrupted source meshes [PITH_FULL_IMAGE:figures/full_fig_p010_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Physically inaccurate samples. Our model generates plausible deformations for the elastic simulation experiment, however can produce inaccuracies that are physically impossible, e.g., the penetration into the ground plane on the right (seen from underneath), as well as self-penetration. iid Gaussian noise on the vertices [PITH_FULL_IMAGE:figures/full_fig_p011_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison to state of the art in point cloud generation, TIGER [Ren et al. 2024b]. By using a triangulation-agnostic intrinsic network over a mesh, we are able to preserve and produce minute details such as the face and individual fingers; these are completely gone from the point cloud generated by TIGER. E Comparison to Point Cloud Generation Different 3D representations serve different goals (e.g., vox… view at source ↗
read the original abstract

This paper tackles the task of learning to generate signals over triangle meshes in a triangulation-agnostic manner, meaning the trained model can be applied to different meshes and triangulations effectively. Practically, the paper adapts the flow matching (FM) paradigm to a mesh-based, triangulation-agnostic setting. Theoretically, it proposes a specific noise distribution which is triangulation agnostic, to be used inside the FM model's denoising process. While noise distributions are usually trivial to devise for, e.g., images, devising a triangulation-agnostic distribution proves to be a much more difficult task. We formulate a mathematical definition of triangulation agnosticism of distributions, via their spectrum. We then show that a discretization of a specific Gaussian random field called a Mat\'ern process holds these desired properties, and provides a simple and efficient sampling algorithm. We use it as our noise model, and adapt FM to the triangulation-agnostic setting by using a state-of-the-art approach for learning signals on meshes in the gradient domain -- PoissonNet -- as the denoiser. We conduct experiments on elaborate tasks such as sampling elastic rest states, and generating poses of humanoids. Our method is shown to be capable of producing highly realistic results for meshes of over one million triangles, significantly exceeding the state-of-the-art in quality and diversity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a triangulation-agnostic adaptation of flow matching for generating signals on triangle meshes. It defines triangulation agnosticism of a distribution via its spectrum, shows that a discretization of the Matérn Gaussian random field satisfies this definition and admits an efficient sampling procedure, and integrates the resulting noise model with PoissonNet (operating in the gradient domain) as the denoiser. Experiments are reported on sampling elastic rest states and generating humanoid poses, with claims of high realism and diversity on meshes exceeding one million triangles.

Significance. If the spectral definition is shown to be sufficient for full trajectory invariance and the experimental results are reproducible, the work would address a practically important limitation in mesh-based generative modeling by allowing a single trained model to be applied to meshes with arbitrary triangulations. The combination of Matérn noise with PoissonNet and flow matching is a targeted technical contribution that could influence downstream applications in geometry processing.

major comments (2)
  1. [Abstract] Abstract and theoretical sections: the claim that the Matérn discretization satisfies the spectrum-based definition of triangulation agnosticism is stated without derivation details, error bounds, or explicit verification that the discrete spectrum matches the continuous Matérn spectrum under arbitrary triangulations; this is load-bearing for the central claim that the noise model is triangulation-agnostic.
  2. [Theoretical contribution] The definition of triangulation agnosticism is restricted to spectral properties of the marginal noise distribution. It remains to be shown that this property propagates through the full flow-matching trajectory (iterative denoising with PoissonNet in the gradient domain), because the denoiser may couple to mesh-dependent discrete gradient and mass-matrix operators that are not controlled by the noise spectrum alone.
minor comments (1)
  1. The sampling algorithm for the Matérn discretization is described as simple and efficient; adding pseudocode or a short complexity analysis would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the careful reading and insightful comments. We address each major comment below and indicate the revisions we intend to make to strengthen the theoretical presentation while remaining faithful to the manuscript's contributions.

read point-by-point responses

  1. Referee: [Abstract] Abstract and theoretical sections: the claim that the Matérn discretization satisfies the spectrum-based definition of triangulation agnosticism is stated without derivation details, error bounds, or explicit verification that the discrete spectrum matches the continuous Matérn spectrum under arbitrary triangulations; this is load-bearing for the central claim that the noise model is triangulation-agnostic.

    Authors: We agree that additional derivation details and verification would improve clarity. The theoretical section derives the discrete Matérn field from the eigen-decomposition of the mesh Laplacian and shows spectral consistency in the continuum limit. In the revision we will expand this with explicit intermediate steps, include approximation error bounds from finite-element analysis of the Laplace-Beltrami operator, and add numerical plots comparing discrete and continuous spectra on multiple triangulations of the same geometry. revision: yes

  2. Referee: [Theoretical contribution] The definition of triangulation agnosticism is restricted to spectral properties of the marginal noise distribution. It remains to be shown that this property propagates through the full flow-matching trajectory (iterative denoising with PoissonNet in the gradient domain), because the denoiser may couple to mesh-dependent discrete gradient and mass-matrix operators that are not controlled by the noise spectrum alone.

    Authors: This observation correctly identifies the scope of our definition. The flow-matching construction transports the marginal noise distribution to the data distribution, and PoissonNet operates in the gradient domain to reduce sensitivity to specific discretizations. Our experiments, including successful transfer to meshes exceeding one million triangles with varying triangulations, provide empirical evidence that the learned model generalizes. A rigorous proof that the entire ODE trajectory remains invariant under arbitrary re-triangulations would require additional analysis of the learned vector field and is not fully established in the current work; we will add a dedicated discussion paragraph acknowledging this limitation and summarizing the supporting experiments. revision: partial

Circularity Check

0 steps flagged

No circularity: definition and verification of Matérn noise properties are independent

full rationale

The paper introduces an original mathematical definition of triangulation agnosticism based on the spectrum of the distribution. It then constructs and verifies that a discretization of the Matérn Gaussian random field satisfies this definition while also yielding an efficient sampling procedure. This constitutes a standard constructive argument rather than any reduction of the claimed result to a fitted parameter, renamed input, or self-referential equation. The flow-matching adaptation relies on the external PoissonNet denoiser operating in the gradient domain, which is treated as an independent component. No load-bearing derivation step collapses by construction to the paper's own prior equations or citations; the central claims retain independent mathematical content and are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central contribution rests on introducing a spectrum-based definition of triangulation agnosticism and verifying that the Matérn discretization satisfies it; no explicit free parameters or new invented entities are described in the abstract.

axioms (1)
  • domain assumption Triangulation agnosticism of a noise distribution can be defined and checked via its spectrum.
    This definition is formulated in the paper as the mathematical foundation for the noise choice.

pith-pipeline@v0.9.0 · 5779 in / 1307 out tokens · 48423 ms · 2026-05-20T02:49:16.785858+00:00 · methodology

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