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Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers

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Pith's one-line read Flow-matching ODE trajectories converge per-sample at terminal time—not just in distribution—even when data lies on a low-dimensional submanifold, with the denoiser as the guiding mechanism.

desk verdict Solid terminal-convergence framework, but Theorem 5.3's rate is false for a valid schedule; deserves review after a fix. read the letter →

arxiv 2412.18730 v4 pith:46AJFWUM submitted 2024-12-25 cs.LG

Zhengchao Wan , Qingsong Wang , Gal Mishne , Yusu Wang This is my paper · ORCID
classification cs.LG
keywords flowmatchingper-trajectoryconvergencedenoiserattractingandabsorbingdynamicsdatageometrysubmanifoldsupportmemorizationequivarianceofmaps
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

Flow matching (FM) is a generative framework that samples by integrating a learned vector field, but what individual trajectories do at the end of the process—whether they actually land on the data—has been open, especially when the data sits on a low-dimensional submanifold. This paper proves that under mild assumptions (finite second moment, positive reach, and a local mass lower bound on the support), the flow map limit $\Psi_1(x)$ exists for almost every starting point, is measurable, pushes the Gaussian prior exactly onto the data distribution, and is approached at rate $O(\sigma_t^{\zeta/2})$ for any fixed $0 < \zeta < 1$. The same attracting-and-absorbing mechanism yields a three-stage description of sampling: trajectories first drift toward the data mean, then are drawn into local clusters, and finally are absorbed onto the data support. Per-trajectory convergence is what one-step distillation and consistency-style models implicitly require, and the terminal analysis also yields quantitative statements about memorization and about how flows transform under rotations, scalings, and translations. The paper's central object is the denoiser—the conditional mean of the data given a noisy observation—which turns out to be the only data-dependent part of the dynamics.

What carries the argument

The engine is the denoiser $m_\sigma(x) = E[X \mid X_\sigma = x]$—the conditional mean of the data $X$ given a noisy observation $X_\sigma = X + \sigma Z$—which is the only data-dependent ingredient of the FM vector field. Reparameterizing the ODE by the noise-to-signal ratio $\sigma = \beta_t/\alpha_t$ puts the dynamics in the form $dx_\sigma/d\sigma = -(1/\sigma)(m_\sigma(x_\sigma) - x_\sigma)$, so the trajectory always points toward the current denoiser value. Two meta-theorems convert properties of the denoiser into trajectory control: an attracting theorem and an absorbing theorem, both driven by the acute-angle condition $\langle m_\sigma(x) - x, \mathrm{proj}_\Omega(x) - x\rangle > 0$ against a closed set $\Omega$. For the terminal stage the key input is a posterior-concentration estimate giving $\|m_\sigma(z) - \mathrm{proj}_\Omega(z)\| = O(\sigma^\zeta)$ uniformly near the support (with rate $\sigma\sqrt{m}$ on an $m$-dimensional manifold and exponential decay for finite discrete data), which feeds an absorbing argument that keeps trajectories inside bounded tubes around the support and away from the medial axis, where the denoiser's limit is discontinuous. These rates make the $1/\sigma$ drift integrable, which is precisely what turns distributional convergence into per-trajectory convergence.

What would settle it

Two concrete checks settle the boundary of the claim. (1) Take an infinite discrete measure on a compact set with atoms accumulating at one point and weights decaying exponentially in their distance to that point: it has finite 2-moment and positive-reach support but violates Assumption 5.2(2) for every $k$, so integrating the exact FM ODE for it and observing whether trajectories still converge to a single atom, or instead wind near the accumulation point, tests whether the assumption is load-bearing or merely technical. (2) For data uniform on a circle (an $m = 1$ manifold), a high-precision integration measuring the exponent of $\|\Psi_1(x) - \Psi_t(x)\|$ versus $\sigma_t$ distinguishes the theorem's $O(\sqrt{\sigma_t})$ bound from the paper's conjectured $O(\sigma_t)$, pinning down whether the manifold rate is tight.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 5.3: for any data distribution with finite second moment whose support has positive reach and whose local mass does not thin out faster than a power law (Assumption 5.2), the flow map of the FM ODE has a limit $\Psi_1(x) := \lim_{t\to 1} \Psi_t(x)$ for Lebesgue-almost every $x$. The limit is measurable, its pushforward of the Gaussian prior is exactly the data distribution $p$, and the per-trajectory error satisfies $\|\Psi_1(x) - \Psi_t(x)\| = O(\sigma_t^{\zeta/2})$ for any fixed $0 < \zeta < 1$, where $\sigma_t = \beta_t/\alpha_t$ is the noise-to-signal ratio. The rate sharpens to $O(\sqrt{\sigma_t})$ when the data is supported on a smooth submanifold and to $O(\sigma_t)$ for finite discrete measures, and the denoiser $m_\sigma(x) = E[X \mid X_\sigma = x]$ is shown to converge to the projection onto the support for every point outside the medial axis. These results upgrade the previously known distributional convergence of FM sampling to genuine per-trajectory convergence, covering the submanifold case that earlier well-posedness results excluded and providing the terminal map that one-step distillation needs.

Load-bearing premise

The load-bearing premise is Assumption 5.2(2): near every point of the data support, the probability mass inside a ball of radius $r$ must be at least $C_R r^k$ for some $k \ge 0$, and the terminal-convergence proof needs this uniform lower bound to turn posterior concentration into the denoiser rate that makes the $1/\sigma$ drift integrable; the paper's remark that this holds for any discrete distribution is only correct for finite discrete measures, not for infinite ones with accumulating atoms.

Editorial extensions

If this is right

  • Per-trajectory terminal convergence supplies the missing object for one-step distillation: the limit map $\Psi_1$ that consistency-style models train against is proved to exist, be measurable, and exactly transport the prior to the data.
  • The $O(\sigma_t^{\zeta/2})$, $O(\sqrt{\sigma_t})$, and $O(\sigma_t)$ rates mean trajectories barely move in the terminal stage, so sampling steps spent there are nearly wasted; the paper's practical corollary is to shift integration effort toward the intermediate stage where the denoiser changes most.
  • For finite discrete (empirical) data, every training point $x_i$ attracts trajectories inside its shrunk Voronoi cell $V_i^\epsilon$, with per-point attraction strength $\sigma_0(V_i^\epsilon)$ that grows with the point's weight and isolation—which explains why duplicated or isolated training samples are the ones a trained model memorizes.
  • An asymptotically optimal denoiser—even one severely corrupted at intermediate noise levels—still drives every trajectory to a training point, so avoiding memorization requires the near-terminal denoiser itself to generalize, that is, to approximate projection onto the true data manifold rather than onto training points.
  • The flow map $\Psi_1$ commutes with rotations, scalings, and translations of the ambient space, so a model trained on a transformed dataset is exactly the transformed flow of the original model; this makes augmentation-based reasoning about flows safe and permits training in a low-dimensional affine subspace and then lifting the result.

Reading between the lines

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

  • The local mass lower bound (Assumption 5.2(2)) is the true crux: the paper's remark that 'any discrete distribution' satisfies it with $k = 0$ holds for finite discrete measures but not for infinite ones with accumulating atoms, so terminal convergence for such 'sticky' supports is neither proved nor disproved here and is a natural stress test of the theorem's boundary.
  • Because the rates are stated for the exact, unlearned denoiser, a trained network whose near-terminal error shrinks slower than $O(\sigma^\zeta)$ falls outside the absorbing argument; this suggests a training diagnostic—monitoring $\|m_\sigma^\theta(x_\sigma) - x_\sigma\|$ through the final $O(\sqrt{\sigma_t})$ window—as a predictor of whether sampling will land on or off the data support.
  • The three-stage picture implies a scheduling principle the paper only gestures at: estimate $\sigma_{\mathrm{init}}$ and $\sigma_{\mathrm{cluster}}$ from dataset quantities (diameter, cluster separation, cluster weights) and allocate integration steps in proportion to denoiser change; because the proven constants are worst-case, the stage boundaries are better treated as tunable hyperparameters th
  • If the conjectured $O(\sigma_t)$ manifold rate is correct, the terminal window shrinks further, strengthening the case for early stopping and coarse final schedules; a closed-form computation on the $m$-sphere with uniform density would likely settle the conjecture before a general proof.
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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 paper analyzes flow-matching (FM) ODE trajectories through the denoiser, introducing a noise-to-signal ratio parametrization and a pair of meta-theorems for attracting and absorbing dynamics. It characterizes three stages of ODE evolution: an initial approach to the data mean, an intermediate attraction to local clusters, and a terminal phase in which trajectories converge to the data support. The main claims include well-posedness of the FM ODE on [0,1) under a finite second moment, terminal convergence for data supported on submanifolds, convergence-rate bounds, flow-map equivariance under similarity transformations, and a memorization analysis for discrete empirical measures. Proofs are deferred to a detailed appendix, and experiments on synthetic clusters, CIFAR-10, and FFHQ illustrate the predicted stage behavior.

Significance. If the stated results hold, this would be a substantial contribution: it provides the first per-trajectory terminal convergence analysis for FM ODEs with submanifold-supported data, a clean schedule-independent parametrization via σ, explicit rates for manifold and discrete cases, and falsifiable predictions about memorization and sampling-stage allocation. The paper also ships detailed appendix proofs, explicit tractable examples (e.g., Gaussian subspaces), and experiments that use the exact empirical optimal denoiser rather than only fitted neural denoisers. However, the central convergence-rate statement in Theorem 5.3 is false as stated for a legal choice of scheduling functions and a one-point data distribution. The framework is promising, and the existence part of the theorem may survive, but the rate claim and the proof that transfers rates from the σ-coordinate back to the t-coordinate need substantive correction.

major comments (3)
  1. [Section 5.1, Theorem 5.3] The claimed rate ∥Ψ1(x)−Ψ_t(x)∥=O(σ_t^{ζ/2}) for every fixed 0<ζ<1 is internally inconsistent with the paper's own setup. Take p=δ_1, α_t=t, β_t=(1−t)^3. These scheduling functions are smooth, strictly monotonic, and satisfy α_0=β_1=0, α_1=β_0=1, so they are admissible in Section 2; p=δ_1 satisfies Assumption 5.2 with k=0. The exact FM ODE solution is x_t=t+(1−t)^3 x_0, so Ψ_1(x_0)−Ψ_t(x_0)=(1−t)(1−(1−t)^2x_0), which is Θ(1−t) for x_0≠0. But σ_t=(1−t)^3/t, so σ_t^{ζ/2}=Θ((1−t)^{3ζ/2}). For any fixed ζ>2/3, the claimed bound fails because (1−t)/σ_t^{ζ/2}→∞. The proof error is visible in Claim 8 of Section I.1: the O( e^{-ζλ/2}) bound is established for the σ-scaled variable z_λ=x_{t(λ)}/α_{t(λ)}, and then transferred to Ψ_t(x) 'up to the change of variable λ→t', but x_t=α_t z_{λ(t)}, so an additional error of order (1−α_t)∥z_∞∥ appears. The corrected statement should include this term, e.g., ∥Ψ_1(x)−Ψ_t(x)∥=O(σ_t^{ζ/2}+1−α_t), or impose a schedule condition such as 1−α_t=O(σ_t^{ζ/2}). The existence part of Theorem 5.3 may still be correct, but the rate statement as written is false.
  2. [Appendix H.1, proof of Theorem 4.1] The proof asserts that the continuity-equation argument of Gao et al. [2024, Theorem 3.1] remains valid verbatim when p is only assumed to have a finite 2-moment and may be singular (e.g., supported on a submanifold). This is a load-bearing step: it is exactly what lets the paper extend well-posedness beyond absolutely continuous data distributions. As written, the assertion is not proved; the standard derivation of ∂_t p_t + ∇·(u_t p_t)=0 typically requires approximation by absolutely continuous measures and a limiting argument to control the flux. Please either provide the full approximation/regularization argument or state the continuity equation as a separately proved lemma with the required assumptions.
  3. [Assumption 5.2] The statement 'Any discrete distribution satisfies the assumptions with k=0' is false for infinite discrete measures with accumulation points. For example, on R take p=∑_{n≥1}2^{-n}δ_{1/n}; then 0∈supp(p), but p(B_r(0)) decays faster than any power of r as r→0, so the local mass lower bound p(B_r(x))≥C_R r^k cannot hold for any k. The subsequent proof of Theorem 5.3 uses this lower bound through Theorem D.4 to obtain the uniform denoiser rate, so the overstatement matters for the stated generality. The theorem is still applicable to finite discrete measures, which cover empirical distributions, but the assumption should be corrected to 'finite discrete distribution' or to an explicit condition on the local mass at accumulation points.
minor comments (4)
  1. [Section 3.2] The phrase 'the acute angel condition' should read 'the acute angle condition'.
  2. [Section 5.1, Theorem 5.3] The statement says 'For p satisfying Theorem 5.2'; the referenced object is an assumption, not a theorem, so it should say 'Assumption 5.2'.
  3. [Section 4.3, Proposition 4.4] The opening line 'With the same assumptions as in Theorem 4.3' refers to Proposition 4.3, not a theorem; the cross-reference should be corrected.
  4. [Section 5.2] The discussion of the CIFAR-10 value σ0(V_i^ϵ)≈0.17 'corresponding to the final quarter of EDM's sampling steps' is schedule-dependent; the specific discretization and noise schedule used for this correspondence should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's derivation chain is self-contained, with theorems proved from stated assumptions rather than reduced to their own inputs.

full rationale

The central claims are derived, not assumed: the denoiser mσ(x) is defined directly from the data measure p via the posterior conditional mean (Eqs. 9 and 12), and the vector field is expressed in terms of this denoiser by direct computation (Eq. 11). The attracting and absorbing meta-theorems (Theorems C.7, 3.2, C.11) are conditional statements whose hypotheses are subsequently established by independent posterior-concentration and denoiser-rate estimates (Theorems D.1, D.4, D.5, D.7), under the explicit mass lower bound in Assumption 5.2. Theorem 5.3 then combines these lemmas with a concentration estimate for qλ to show the trajectory is Cauchy and hence converges; this is a deduction from the stated assumptions, not a fit or a renaming. The stage boundaries σinit, σ0(S, ε), and σ0(V_i^ε) are computed from data-dependent quantities such as support diameter, cluster weight, and separation, and are used to validate predicted qualitative behavior; they are not fitted to observed trajectories and then called predictions. No load-bearing self-citation, imported uniqueness theorem, or ansatz smuggled in by citation appears: prior work such as Pidstrigach [2022] and Permenter and Yuan [2024] is used only for context or comparison, not to justify the paper's convergence claims. The skeptic attack on the schedule-dependence of the O(σ_t^{ζ/2}) rate concerns correctness of a quantitative bound under specific scheduling functions, not circularity: even if the rate statement needed additional schedule assumptions, that would not make the result definitionally equivalent to its inputs. Overall, the derivation is self-contained and no circular step was found.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proofs introduce no fitted parameters; constants such as ζ, ϵ, and δ are arbitrary user-chosen quantities absorbed into the theorems, and the σ0 values in the memorization discussion are computed from the data distribution rather than fit to observed trajectories. The central claim rests on the standard FM construction (Gaussian prior, independent X and Z, smooth schedules), finite 2-moment for well-posedness, and the regularity Assumption 5.2 for terminal convergence. The Local Cluster Assumption is an ad hoc condition used only for the intermediate-stage result.

assumptions (7)
  • domain assumption The prior p_prior is the standard Gaussian N(0,I), and X and Z are independent in the construction of the probability path p_t.
    Section 2.1 and the Discussion; all theorems are stated for this standard FM setup, and dependent couplings are explicitly left to future work.
  • domain assumption Scheduling functions α_t and β_t are smooth on [0,1] with α0=β1=0 and α1=β0=1, and are monotonic.
    Section 2.1; needed for the noise-to-signal σ reparametrization to be invertible and for endpoint behavior at t=0 and t=1.
  • domain assumption The data distribution p has a finite 2-moment.
    Used for well-posedness (Theorem 4.1), posterior existence (Proposition C.1), and the concentration estimates in Section D.
  • domain assumption Assumption 5.2: positive reach of supp(p) and the local lower mass bound p(B_r(x)) ≥ C_R r^k on compact pieces of the support.
    Load-bearing premise for terminal convergence Theorem 5.3, and used for the uniform denoiser-to-projection rate in Claim 4 and the absorbing property in Theorem C.11.
  • ad hoc to paper Local Cluster Assumption: a cluster S with diam(S)=D and d_conv(S)(x)>2D for all x∈supp(p)\S.
    Used only for the intermediate-stage Theorems 4.3 and 4.4; the paper acknowledges that real datasets seldom satisfy it exactly and provides a partial extension via Gaussian convolution.
  • domain assumption For the refined manifold rate, M is a closed m-dimensional submanifold with positive reach, smooth nonvanishing density, and bounded second fundamental form up to first derivatives.
    Theorem 5.4 and Appendix I.2; needed for the coefficient-of-variation style volume growth bound in Lemma I.1 and the Taylor expansion of the exponential map.
  • standard math Standard results from convex geometry, reach theory, and continuity equations are used without reproof.
    Appendices B and H rely on Federer reach properties, supporting hyperplanes, and Ambrosio et al. continuity equation theory.

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Pith. "Pith review of Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers." pith.science (2026). https://pith.science/paper/46AJFWUM

@misc{pith2026241218730,
  author       = {Pith},
  title        = {Pith review of: Elucidating Flow Matching ODE Dynamics with Respect to Data Geometries and Denoisers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/46AJFWUM}},
  note         = {Machine review of arXiv:2412.18730}
}
read the original abstract

Flow matching (FM) models extend ODE sampler based diffusion models into a general framework, significantly reducing sampling steps through learned vector fields. However, the theoretical understanding of FM models, particularly how their sample trajectories interact with underlying data geometry, remains underexplored. A rigorous theoretical analysis of FM ODE is essential for sample quality, stability, and broader applicability. In this paper, we advance the theory of FM models through a comprehensive analysis of sample trajectories. Central to our theory is the discovery that the denoiser, a key component of FM models, guides ODE dynamics through attracting and absorbing behaviors that adapt to the data geometry. We identify and analyze the three stages of ODE evolution: in the initial and intermediate stages, trajectories move toward the mean and local clusters of the data. At the terminal stage, we rigorously establish the convergence of FM ODE under weak assumptions, addressing scenarios where the data lie on a low-dimensional submanifold-cases that previous results could not handle. Our terminal stage analysis offers insights into the memorization phenomenon and establishes equivariance properties of FM ODEs. These findings bridge critical gaps in understanding flow matching models, with practical implications for optimizing sampling strategies and architectures guided by the intrinsic geometry of data.

Figures

Figures reproduced from arXiv: 2412.18730 by the authors.

Figure 1
Figure 1. Comparing two ODE trajectory behaviors. Various versions of the FM model have gained popularity, such as the rectified flow model [Liu et al., 2023], which is utilized in commercial image generation software [Esser et al., 2024]. Furthermore, the succinct and deterministic formulation of the FM model also makes theoretical analysis potentially easier. Despite the empirical success, critical theoretical questions rem… view at source ↗
Figure 2
Figure 2. Three stages of an FM ODE trajectory with synthetic data. The curve with blue progression shows an FM ODE trajectory, with an arrow indicating direction. The shaded region indicates the convex hull of data. Three stages are visible: initially, the trajectory aligns with the data mean (brown point); next, it is attracted to a local cluster (yellow cluster); finally, it converges to a data point (green star). See Sect… view at source ↗
Figure 3
Figure 3. Illustration of the acute angle condition. With some quantitative bound on the acute angle condition above, the ODE trajectory will be attracted toward Ω. Theorem 3.1 (Informal - Attracting toward sets). Let (xσ)σ∈(σ2,σ1] be an ODE trajectory of Equa￾tion (11) starting from some xσ1 . Assume that the trajectory avoids the medial axis ΣΩ of some closed Ω and satisfies the acute angle condition in a quantitative manne… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Roadmap of theoretical results. Our analysis builds upon three foundational compo￾nents: (1) well-posedness of FM ODEs in [0, 1), establishing existence and uniqueness of trajecto￾ries—setting the foundation for subsequent analysis, (2) attracting and absorbing propert…
Figure 5
Figure 5. Figure 5: The denoiser mσ(x) for the two point example with various σ. The discontinuity behavior can be illustrated by the following simple example of a two-point data distribution which can be easily extended to higher dimensions. Example C.5. Let p = 1 2 δ−1+ 1 2 δ1 be a prob…
Figure 6
Figure 6. Figure 6: The synthetic dataset showing three clusters with different scales and sample sizes. The brown point marks the data mean. We generate a synthetic dataset in R 2 consisting of three distinct clusters shown in [PITH_FULL_IMAGE:figures/full_fig_p071_6.png]
Figure 7
Figure 7. Figure 7: Evolution of a single trajectory showing three distinct stages. Initial stage: Starting with σ1 = 100 and initial distance to mean 122, Theorem 4.2 predicts σinit = 13 with ζ = 0.5. This suggests that the trajectory will approach the mean at least before σinit = 13. We…
Figure 8
Figure 8. Figure 8: Multiple trajectories from different initial samples. Left: Global view showing roughly linear paths to mean. Middle: Zoomed view. Right: Close view near data support. Blue crosses mark xσinit for each path. Convex Hull Cluster 1 Cluster 2 Cluster 3 Data mean _cluster …
Figure 9
Figure 9. Figure 9: Detailed view of intermediate and terminal stages for two example trajectories. 73 [PITH_FULL_IMAGE:figures/full_fig_p073_9.png]
Figure 10
Figure 10. Figure 10: Memorization behavior with asymptotically optimal denoiser. Left: Global view showing trajectories under perturbation, with perturbation vector displayed. Middle: Zoomed view of trajectories. Right: Close view near data support. While perturbations initially steer tra…
Figure 11
Figure 11. Figure 11: Initial mean attraction in CIFAR-10 trajectories. Results averaged over 10,000 random initializations with shaded regions showing ±1 standard deviation. Both empirical optimal and trained denoisers exhibit strong initial attraction to the dataset mean when σ is large,…
Figure 12
Figure 12. Figure 12: Reference case: Sample generation with unperturbed empirically optimal denoiser. Top: ODE trajectory xσ. Bottom: Denoiser outputs mσ(xσ). Note the clear image structure throughout and smooth convergence to a training image. 76 [PITH_FULL_IMAGE:figures/full_fig_p076_12.png]
Figure 13
Figure 13. Figure 13: Perturbed case: Sample generation with heavily perturbed but asymptotically optimal denoiser. Despite the denoiser outputs being severely corrupted by noise (bottom) compared to the reference case, the ODE trajectory (top) still remarkably converges to a training imag…
Figure 14
Figure 14. Figure 14: Quantitative result of convergence and memorization. Evolution of distances to the CIFAR-10 training set for both trajectories and denoiser outputs using the empirical optimal denoiser and its perturbed version, averaged over 10,000 random seeds. Left: Reference case …
Figure 15
Figure 15. Figure 15: t-SNE Visualization of FFHQ (Human Face) Dataset. We downsample the original dataset from high-resolution (1024×1024) human face images to 64×64 resolution—aligning the training procedure in EDM and subsample 10,000 data points to perform t-SNE. The visualization show…
Figure 16
Figure 16. Figure 16: Illumination-based Absorption Behavior. Samples generated using three initializa￾tion strategies: random noise (left), noise near dark illumination regions (middle), and noise near bright illumination regions (right). Random initialization produces samples across the …

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · cited by 2 Pith papers

  1. [1]

    For eachx∈R d, there exists a unique pointprojK(x) ∈K such that ∥x−proj K(x)∥ = dK(x)

    work page

  2. [2]

    The distance functiondK(x)is a convex function

    work page

  3. [3]

    For any r≥ 0, the r-thickening of K, defined asBr(K) := {x∈R d : dK(x) < r}, is a convex set

    work page

  4. [4]

    The diameter ofΩis the same as the diameter of its convex hull, that is diam(Ω) = diam(conv(Ω))

    work page

  5. [5]

    memorize

    Leta >0, then a setΩis convex if and only ifa·Ωis convex. 23 B.2 Metric Geometry Notions and Results LetΩ ⊂R d be a closed subset. Recall thatΣΩ denotes the medial axis ofΩand dΩ : Rd →R is defined by dΩ(x) := inf y∈Ω ∥x−y∥ . We now consider certain properties of the projection function projΩ : Σc Ω →Ω, whereΣ c Ω :=R d\ΣΩ. We first recall the definition ...

    work page 1998

  6. [6]

    For anyt2 > t1 we have that yt2 ≤e −k(t2−t1)yt1 + Z t2 t1 e−k(t2−t)ϕ(t)dt

    work page

  7. [7]

    Proof of Theorem C.6.By multiplying the integrating factorekt, we have that d ektyt dt =e kt d yt dt +ke ktyt ≤ϕ(t)e kt

    Iflim t→∞ ϕ(t) = 0, thenlim t→∞ yt = 0. Proof of Theorem C.6.By multiplying the integrating factorekt, we have that d ektyt dt =e kt d yt dt +ke ktyt ≤ϕ(t)e kt. Then for allt2 > t1, we have ekt2yt2 ≤e kt1yt1 + Z t2 t1 ektϕ(t)dt, yt2 ≤e −k(t2−t1)yt1 + Z t2 t1 e−k(t2−t)ϕ(t)dt. 32 This proves Item 1. For Item 2, we will first show thatyt is bounded. As ϕ(t)d...

    work page

  8. [8]

    If ⟨mσ(xσ) −proj Ω(xσ),proj Ω(xσ) −x σ⟩ ≤ζ∥xσ −proj Ω(xσ)∥2 for some0 ≤ζ < 1along the trajectory, thend Ω(xσ)decreases along the trajectory with rate: dΩ(xσ)≤ σ1−ζ σ1−ζ 1 dΩ(xσ1), In particular, ifσ2 = 0, thend Ω(xσ)is guaranteed to converge to zero asσ→0

    work page

Show all 23 references
  1. [9]

    If σ2 = 0and |⟨mσ(xσ) −proj Ω(xσ),proj Ω(xσ) −x σ⟩| ≤ϕ(σ)for some function ϕ(σ)along the trajectory withlim σ→0 ϕ(σ) = 0, then lim σ→0 dΩ(xσ) = 0. RemarkC.8.In fact, when considering the parameter λ = −log (σ)and the trajectory zλ := xσ(λ), we obtain the following convergence ...

  2. [10]

    For any r > 0, if mσ(x) ∈K for any x∈∂B r(K)and any σ∈ (σ2, σ1], then Br(K)is absorbing for(x σ)σ∈(σ2,σ1]

  3. [11]

    35 Proof of Theorem C.9.For the first item, for anyx∈∂B r(K), since mσ(x)lies in the convex set K, we can apply Item 2 of Theorem B.9 to conclude ⟨mσ(x)−proj K(x),proj K(x)−x⟩ ≥0

    If the interiorK◦ of K is not empty andmσ(x) ∈K ◦ for any x∈∂K and any σ∈ (σ2, σ1], thenKis absorbing for(x σ)σ∈(σ2,σ1]. 35 Proof of Theorem C.9.For the first item, for anyx∈∂B r(K), since mσ(x)lies in the convex set K, we can apply Item 2 of Theorem B.9 to conclude ⟨mσ(x)−pro...

  4. [12]

    Ifx σ1 ∈conv(supp(p b)), thenx σ ∈conv(supp(p b))for anyσ∈(0, σ 1]

  5. [13]

    If xσ1 /∈conv(supp(pb)), then xσ moves toward conv(supp(pb))with the following decay guarantee: dconv(supp(pb))(xσ)≤ √ σ2 +δ 2 p σ2 1 +δ 2 dconv(supp(pb))(xσ1), for anyσ∈(0, σ 1]. Proof of Theorem C.13.This is a direct consequence of Theorem 3.4 and Theorem C.12 D Denoiser Ana...

    2024

  6. [14]

    Consider the transformationu = σz

    So, we can chooser0 small enough at the beginning so that there exists some constantC3 >0for allu∈U r0, we have that uT (I−II v)u+R 2(u)≥λ min∥u∥2 −C 2∥u∥3 > C3∥u∥2. Consider the transformationu = σz. Then, we letσ′ := 2r0(C2σ) 1 3 which goes to0as σ→ 0 and for sufficiently sm...

  7. [15]

    For each x1 ∈R m, there is ∥xt −α tx1∥2 = ∥xt −α tx1∥2 + ∥yt∥2 where xt = ( xt, yt)and x1 = (x1,0)

  8. [16]

    The denoiser mt always lies in the subspaceRm since the data distributionp is supported on Rm. These two observations imply that the FM ODE can be decoupled into two ODEs:    dxt dt = ˙βt βt xt + ˙αtβt −α t ˙βt βt Z exp − ∥xt−αtx1∥2 2β2 t x1 R exp − ∥xt−αtx′ 1∥2 2β2...

  9. [17]

    , ≤(diam(Ω)) 2 R (Ω\S) exp − 1 2σ2 ∥x−x 1∥2 p(dx1) R S exp − 1 2σ2 ∥x−x 1∥2 p(dx1) , By the assumption thatdconv(S)(x)≤D/2−ϵ, we have that for allx 1 ∈S ∥x−x 1∥ ≤D/2−ϵ+D= 3D/2−ϵ, and for allx1 ∈Ω\S ∥x−x 1∥ ≥2D−(D/2−ϵ) = 3D/2 +ϵ. We then have that dW,2 ησ, ηS σ 2 ≤(diam(Ω)) 2 R...

  10. [18]

    For the second part, we useλ = −logσ as the reparametrization and the corresponding trajectory zλ := xσ(λ)

    This concludes the proof of the first part. For the second part, we useλ = −logσ as the reparametrization and the corresponding trajectory zλ := xσ(λ). We want to show that the distancedconv(S)(zλ)goes to zero as λ goes to infinity. By taking the derivative ofd2 conv(S)(zλ), a...

    2008

  11. [19]

    for anyt 1 ∈[0,1)and any compact setK⊂R d, Z t1 0 sup K ∥vt∥+ Lip(v t, K) <∞, whereLip(v t, K)is the Lipschitz constant ofvt onK

  12. [20]

    Then, the ODE dx dt =u t(x), x(0) =x0 has a unique solutionx(t)for all t∈ [0, 1)

    for any0< t 0 < t1 <1, there exists a constantCt0,t1 such that ∥vt(x)∥ ≤Ct0,t1(1 +∥x∥),for allt∈[t 0, t1]. Then, the ODE dx dt =u t(x), x(0) =x0 has a unique solutionx(t)for all t∈ [0, 1). Furthermore, the flow mapΨt : Rd →R d is continuous and satisfies that(Ψt)#q0 =q t for a...

    2008

  13. [21]

    turning points

    the convergence rate is given by∥zλ −z λδ ∥=O(e − ζλ 2 ). Proof of Claim 7.For anyδ >0, by Claim 5 and Claim 6, there exist large enoughλδ and Rδ, such that •with probability at least1−δ, we have thatz λδ ∼q λδ lies inB Rδ (0)∩B δ(Ω); • the ODE trajectory(zλ)λ∈[λδ,∞) starting ...

    1990

  14. [22]

    The distance∥m ∗ σ(x∗ σ)−mean∥between the denoiser output and the dataset mean

  15. [23]

    This ratio can be interpreted as a relative error when approximating the denoiser output with the dataset mean

    The ratio∥m∗ σ(x∗ σ) −mean∥/∥m ∗ σ(x∗ σ) −x ∗ σ∥, which quantifies the distance between the denoiser output and the data mean relative to the trajectory directionm∗ σ(x∗ σ) −x ∗ σ. This ratio can be interpreted as a relative error when approximating the denoiser output with th...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.