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arxiv: 2604.07213 · v2 · pith:BIDBUGJOnew · submitted 2026-04-08 · 💻 cs.LG · math.PR

Diffusion Processes on Implicit Manifolds

Victor Kawasaki-Borruat , Clara Grotehans , Pierre Vandergheynst , Adam Gosztolai This is my paper

Pith reviewed 2026-05-21 09:32 UTC · model grok-4.3

classification 💻 cs.LG math.PR
keywords diffusion processesimplicit manifoldspoint cloudsproximity graphsstochastic differential equationsmanifold learninggenerative modeling
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The pith

Diffusion processes defined on point clouds converge in law to their smooth manifold versions as sampling density increases.

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

This paper develops a method to run diffusion processes that stay on the underlying manifold of high-dimensional data using only scattered point samples. The approach approximates the generator of the diffusion with a graph connecting nearby points and an operator that identifies the local directions of the manifold. The main result shows that these discrete processes approach the true manifold diffusion as more points are added. This matters because it allows sampling and exploring data manifolds without needing explicit maps or projections. The construction supports numerical simulation of paths confined to the data manifold.

Core claim

We introduce Implicit Manifold-valued Diffusions (IMDs) that define stochastic differential equations in the original high-dimensional space whose solutions evolve intrinsically on the underlying manifold. The construction approximates the infinitesimal generator using a proximity graph over the data points and the carré-du-champ operator, which encodes the local tangent spaces and lifts the intrinsic process into ambient coordinates. As the number of samples grows, the discrete diffusion process converges in law on the space of probability paths to its smooth manifold counterpart, and an Euler-Maruyama scheme enables numerical integration.

What carries the argument

A proximity graph over the data points combined with the carré-du-champ operator, which recovers local tangent spaces and lifts the intrinsic diffusion into the ambient high-dimensional coordinates.

If this is right

  • An Euler-Maruyama scheme can be used for numerical integration of the IMDs.
  • In experiments on synthetic manifolds and the MNIST data manifold the simulated paths remain confined to the manifold.
  • The processes enable guided exploration of the data manifold.
  • The framework supplies a foundation for manifold-aware sampling and generative modeling.

Where Pith is reading between the lines

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

  • This construction could be combined with existing generative models to enforce manifold structure during sampling steps.
  • Similar graph-based approximations might extend to other stochastic processes such as jump diffusions on manifolds.
  • Practical tests on datasets with independently verified manifold structure would provide direct checks on convergence rates.

Load-bearing premise

The data points are sampled densely enough from a smooth manifold so that the proximity graph plus carré-du-champ operator accurately recovers the local tangent spaces and the intrinsic generator without additional geometric primitives.

What would settle it

Simulating paths from the discrete IMD on successively denser samples from a known manifold and checking whether the generated probability paths match those of the true intrinsic diffusion, or diverge when the graph fails to capture the geometry.

Figures

Figures reproduced from arXiv: 2604.07213 by Adam Gosztolai, Clara Grotehans, Pierre Vandergheynst, Victor Kawasaki-Borruat.

Figure 1
Figure 1. Figure 1: No prior knowl￾edge of T 2 beyond the sam￾ples (gray) is required to com￾pute the displayed Brownian motion (red). In this work, we take an operator-theoretic approach to diffusion on implicit manifolds. Starting from a proximity graph built from the point cloud XN , we consider the associated random walk graph Laplacian LN ; the discrete generator of a local Markov process on the graph. We then show that,… view at source ↗
Figure 2
Figure 2. Figure 2: Histogram of the endpoint statistic t = ⟨µ, YT ⟩ ∈ [−1, 1] (blue) under Langevin dynamics computed with IMDs, compared with the theoretical density induced by the von Mises–Fisher distribution (red) on S 7 ⊂ R 8 . Close agreement indicates that the simulated process recovers the target equilibrium law. See Section 6.1 for more details. Geometric limits of graph Laplacians This is the theoretical backbone o… view at source ↗
Figure 3
Figure 3. Figure 3: We use the score of a pre-trained SGM sσ(x) ≈ ∇x log pσ(x) as a retraction toward the data manifold. Numerical approximations of IMDs necessarily operate with finite step sizes, whereas the guarantees of Theorems 2 and 3 hold in the infinitesimal limit h → 0. In practice, only applying the Euler￾Maruyama (E-M) discretization of the IMD SDE may lead to off￾manifold deviations due to discretization error, wh… view at source ↗
Figure 4
Figure 4. Figure 4: IMDs produce a smooth transition between two dissimilar data points [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Step size is 1e-3, simulated over 5’000 steps. The Laplacian is computed over 10’000 samples. We notice that while the Swiss Roll is not boundaryless, the nearest-neighbour approach to estimating L(Xt) via P ∗ N prevents off-manifold drift. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of diffusion trajectories (top row) and corresponding radial errors (bottom row). [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of diffusion trajectories on the Swiss roll (top row) and corresponding latent [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
read the original abstract

High-dimensional data are often assumed to lie on lower-dimensional manifolds. We study how to construct diffusion processes on this data manifold using only point cloud samples and without access to charts, projections, or other geometric primitives. Here, we introduce Implicit Manifold-valued Diffusions (IMDs), a data-driven mathematical formalism for defining stochastic differential equations in the original high-dimensional space that describe drifting Brownian particles evolving intrinsically on the underlying manifold. Our construction hinges on approximating the corresponding infinitesimal generator of the diffusion process using a proximity graph over the data and using the carr\'e-du-champ of the generator, which encodes the local tangent spaces of the manifold and lifts the intrinsic process into ambient coordinates. We show that as the number of samples grows, our discrete diffusion process converges in law on the space of probability paths to its smooth manifold counterpart. We further present an Euler-Maruyama scheme for the numerical integration of IMDs. We validate our framework using numerical experiments on synthetic manifolds and the MNIST data manifold, showing that IMDs remain confined over the manifold and enable its guided exploration. Our work provides the mathematical foundation and practical implementations of diffusion processes on data manifolds, opening new avenues for manifold-aware sampling, exploration, and generative modeling.

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

1 major / 2 minor

Summary. The paper introduces Implicit Manifold-valued Diffusions (IMDs), a framework for constructing diffusion processes on an unknown manifold from point-cloud samples alone. It approximates the intrinsic generator via a proximity graph and the carré-du-champ operator, which encodes local tangent spaces, then lifts the process into ambient coordinates. The central theoretical claim is that the resulting discrete process converges in law on path space to the smooth manifold diffusion as the number of samples n tends to infinity. An Euler-Maruyama discretization is provided, and the method is illustrated on synthetic manifolds and the MNIST data manifold, where trajectories remain confined to the manifold and permit guided exploration.

Significance. If the convergence result holds under the stated assumptions, the work supplies a mathematically grounded way to perform intrinsic diffusion on data manifolds without charts or explicit geometric primitives. This could support manifold-aware sampling and generative modeling in high-dimensional settings where only samples are available. The numerical validation on MNIST demonstrates practical confinement to the data manifold, which is a concrete strength of the empirical component.

major comments (1)
  1. [Theorem on convergence in law] Theorem on convergence in law (likely §4 or the main result following the generator construction): the statement claims that the discrete process converges in law to the intrinsic manifold diffusion as n→∞, but does not condition on a scaling regime for the proximity-graph radius ε_n (e.g., ε_n→0 with nε_n^{d+2}→∞ or the analogous k-NN rate). Without this explicit regime, the graph-based generator may converge to the ambient Euclidean Laplacian rather than the intrinsic Laplace-Beltrami operator, undermining the path-space limit to the claimed manifold process. This is load-bearing for the central claim.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction refer to “the space of probability paths” without clarifying whether this is the space of continuous paths equipped with the uniform topology or a weaker Skorokhod-type topology; a brief sentence on the precise function space would improve readability.
  2. [Euler-Maruyama scheme] In the Euler-Maruyama scheme section, the step-size h is introduced without an explicit relation to the graph radius ε_n; adding a short remark on how h should scale with ε_n would clarify the discretization error relative to the graph approximation error.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and valuable comments, which help clarify the conditions for our main convergence result. We address the major comment below.

read point-by-point responses

  1. Referee: [Theorem on convergence in law] Theorem on convergence in law (likely §4 or the main result following the generator construction): the statement claims that the discrete process converges in law to the intrinsic manifold diffusion as n→∞, but does not condition on a scaling regime for the proximity-graph radius ε_n (e.g., ε_n→0 with nε_n^{d+2}→∞ or the analogous k-NN rate). Without this explicit regime, the graph-based generator may converge to the ambient Euclidean Laplacian rather than the intrinsic Laplace-Beltrami operator, undermining the path-space limit to the claimed manifold process. This is load-bearing for the central claim.

    Authors: We agree that an explicit scaling regime for ε_n is necessary to guarantee convergence of the graph Laplacian to the intrinsic Laplace-Beltrami operator. The proof of the main theorem (Section 4) already assumes ε_n → 0 with n ε_n^{d+2} → ∞ (or the analogous k-NN condition) to obtain the required pointwise and uniform convergence of the discrete generator and carré-du-champ operator; these rates are stated in the technical assumptions and used throughout the error analysis. However, the formal theorem statement itself only mentions n → ∞ without restating the ε_n regime. We will revise the theorem to explicitly condition the path-space convergence on this scaling, thereby making the load-bearing assumption transparent. This is a clarification rather than a change to the result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external graph approximation theory

full rationale

The paper defines IMDs by constructing a discrete generator from a proximity graph plus carré-du-champ operator on point-cloud data, then proves that the resulting process converges in law on path space to the intrinsic manifold diffusion as n→∞. This limit statement is an asymptotic consistency result that invokes standard conditions on neighborhood scaling (ε_n→0 with nε_n^{d+2}→∞ or equivalent) drawn from the existing literature on graph Laplacians converging to the Laplace-Beltrami operator. No equation reduces the claimed convergence to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose own justification is internal to the present work. The construction is therefore self-contained against external benchmarks in manifold learning and Dirichlet-form theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the construction implicitly relies on the existence of a smooth manifold from which points are sampled and on the ability of a proximity graph to approximate the Laplace-Beltrami operator and its carré-du-champ. No explicit free parameters or invented entities are named in the provided text.

axioms (1)
  • domain assumption Data points are sampled from a smooth Riemannian manifold embedded in Euclidean space
    Invoked to justify that the proximity graph recovers local tangent spaces and that the discrete process converges to the intrinsic diffusion.

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Forward citations

Cited by 2 Pith papers

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