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REVIEW 4 major objections 5 minor 41 references

Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read By appending one deterministic inertia update to the empirical diffusion ODE, the paper claims, the output distribution matches the true data distribution on a $d$-dimensional $C^2$ manifold in Wasserstein-1 distance at rate…

desk verdict A clean and likely correct resolution of memorization in empirical diffusion models under the manifold hypothesis, held back by small proof typos and an unstated exact-ODE idealization. read the letter →

arxiv 2505.02508 v3 pith:XX4OSIWS submitted 2025-05-05 stat.ML cs.LGmath.STstat.TH

Yang Lyu , Tan Minh Nguyen , Yuchun Qian , Xin T. Tong This is my paper · ORCID
classification stat.MLcs.LGmath.STstat.TH MSC 60H3065Y2058J6562D05
keywords generativemodeldiffusionmemorizationmanifoldhypothesiskerneldensityestimationNadaraya-WatsonestimatorWasserstein-1distancecurseofdimensionality
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

The paper's central claim is that memorization in empirical diffusion models is not an unavoidable consequence of using the empirical score: a single inertia update at the end of the reverse ODE, with no additional training, produces genuinely new samples. Under the manifold hypothesis (data supported on a compact $C^2$ $d$-dimensional manifold with density bounded away from zero), the output distribution of this inertial diffusion model (IDM) is within Wasserstein-1 distance $O(n^{-2/(d+4)})$ of the true distribution, with high probability and constants independent of the ambient dimension $D$. Because the memorized empirical distribution is only $\widetilde{O}(n^{-1/d})$ close to the truth, the IDM bound is strictly better once $d\ge 5$, which is the formal sense in which the model provably generates novel points rather than copying training data. The proof shows IDM samples behave like Gaussian kernel density estimates on the manifold, giving a quantitative bridge between diffusion models and manifold learning.

What carries the argument

The central object is the inertia update, one deterministic correction applied after running the empirical diffusion ODE up to time $T-h^2$. Algebraically, the update is an application of the Nadaraya--Watson estimator---a kernel-weighted average of the training points---to a noisy point $X_U+\sigma\xi$, with bandwidth $\sigma\asymp h$. The argument rests on three properties proved for this object: it truncates the normal (off-manifold) part of the noise and moves the point along a geodesic by its tangent component; its samples match a manifold Gaussian kernel density estimate within $O(h^2)$ in Wasserstein-1 distance; and the manifold KDE is $O(n^{-2/(d+4)})$ close to $p_0$ in sup norm with high probability.

What would settle it

Take $n$ points drawn from a known $d$-dimensional manifold embedded in $\mathbb{R}^D$ with $D$ large, run Algorithm 1 with $h=n^{-1/(d+4)}$ and a high-accuracy ODE solver, and estimate $W_1$ between IDM output and the true distribution. If the empirical slope is not $n^{-2/(d+4)}$ or the error grows with $D$, the central bound fails; equally, adding a small isotropic ambient component (so data are no longer exactly on a manifold) should break the independence-of-$D$ claim and the predicted slope.

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Extended reading notes

Core claim

The core discovery is that the empirical score function---the exact object that makes the reverse process collapse onto training points---already carries enough geometric information to synthesize new data, provided the reverse ODE is stopped before the final singularity and the last interval is replaced by the inertia update $\widehat{Z}_T = \alpha_{h^2}^{-1}(Z_{T-h^2}+\sigma_{h^2}^2 \nabla \log \hat{p}_{T-h^2}(Z_{T-h^2}))$. The paper proves (Theorem 2.5) that with $h=n^{-1/(d+4)}$ and Assumption 2.1, the distribution $\hat{p}_{\mathrm{IDM}}$ produced by Algorithm 1 satisfies $W_1(\hat{p}_{\mathrm{IDM}}, p_0) \le O(d^2 k^2 n^{-2/(d+4)})$ with probability at least $1-n^{-k}$, whether the initial point is drawn from $\hat{p}_T$ or from the standard normal once $T$ is large. The same chain of arguments shows $\hat{p}_{\mathrm{IDM}}$ is within $O(h^2)$ of the Gaussian kernel density estimator on the manifold; this is why the rate matches the minimax $C^2$ density-estimation rate and why the ambient dimension $D$ appears nowhere in the bound.

Load-bearing premise

The argument stands on the premise that the data really do lie on a smooth $d$-dimensional surface with density bounded away from zero, and that the reverse ODE is integrated exactly; if the data spread off that surface or the numerical solver introduces error, the claimed rate is not guaranteed to hold.

Editorial extensions

If this is right

  • If the theorem is right, empirical diffusion models with the inertia update attain the minimax $C^2$ density-estimation rate $O(n^{-2/(d+4)})$ with no neural-network training, so any full-capacity score approximation inherits a memorization-free baseline.
  • The error bound is independent of the ambient dimension $D$ and requires only a sample size above a threshold independent of $D$, so the method avoids the curse of dimensionality under the manifold hypothesis.
  • For intrinsic dimension $d \ge 5$, the IDM distance to the true distribution shrinks faster than the memorization rate $W_1(\hat{p}_0,p_0)=\widetilde{O}(n^{-1/d})$, which is the formal sense in which the model creates new samples rather than copying.
  • IDM is a generative counterpart to manifold kernel density estimation: it produces samples from a distribution within $O(h^2)$ of the manifold KDE without requiring a chart of the manifold, at $O(n)$ cost per sample.

Reading between the lines

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

  • The same inertia step should work as a standalone high-dimensional manifold denoiser: Theorem 2.2's normal-truncation property is independent of the ODE context, so the estimator could be applied directly to project noise-perturbed points back to an unknown manifold.
  • A natural extension is to test whether the rate survives coarse numerical integration; since the theorem assumes exact ODE evolution, simulations with low-order solvers could reveal the practical step-size regime.
  • Because the IDM distribution is within $O(h^2)$ of a manifold KDE, bandwidth-selection rules from nonparametric statistics could be imported to choose $h$ automatically without changing the minimax rate.
  • If the manifold and density were smoother than $C^2$, the Gaussian-kernel structure of the update would still cap the rate at the $C^2$ minimax exponent, suggesting that a higher-order kernel variant could improve the exponent.
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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

4 major / 5 minor

Summary. The paper proposes 'inertia diffusion models' (IDM), a training-free modification of the empirical diffusion model in which the final segment of the reverse ODE is replaced by a single Nadaraya–Watson estimator-based update (Algorithm 1). Under the assumption that the data lie on a compact C^2 d-dimensional manifold embedded in R^D, the authors claim that the output distribution of IDM is within Wasserstein-1 distance O(d^2 k^2 n^{-2/(d+4)}) of the true data distribution with high probability, independently of the ambient dimension D. The proof strategy is: (i) express the empirical score update as a Nadaraya–Watson estimator, (ii) show this estimator approximately projects onto the manifold while preserving the tangent component, (iii) compare the resulting distribution to a manifold kernel density estimator, and (iv) use standard KDE concentration and optimal-transport inequalities. Numerical experiments on SO(4) embedded in R^D support the claimed scaling in n and D.

Significance. If the central claim holds, this is a valuable result: it provides a principled, training-free mechanism for avoiding memorization in empirical diffusion models, achieves the minimax manifold-KDE rate n^{-2/(d+4)}, and is independent of the ambient dimension, which is a strong and practically relevant guarantee. The paper also establishes a clean conceptual connection between diffusion models and manifold kernel density estimation, and the NWE-projection mechanism is an interesting contribution in its own right. The analysis is anchored to standard external results (Bernstein's inequality, KDE bias-variance analysis, Villani's W1-L∞ bound, Fournier–Guillin empirical measure rates) and the bandwidth h = n^{-1/(d+4)} is the standard minimax choice rather than a fitted parameter. That said, the paper in its current form contains several load-bearing gaps: the theorem analyzes an exact ODE flow while Algorithm 1 requires a numerical solver, and several displayed probability bounds and union-bound steps do not close as written. These issues are correctable, but they must be fixed before the advertised claim is rigorously established.

major comments (4)
  1. [Algorithm 1 and Theorem 2.5] Algorithm 1 instructs the user to 'solve ODE d/dt Z_t = Z_t + ∇log p̂_{T-t}(Z_t) up to T-h^2', but Theorem 2.5 and its proof analyze the exact flow of this ODE. No discretization error is bounded for any numerical integrator. As a result, the main theorem does not certify the distribution produced by any actual numerical implementation of Algorithm 1. This is a load-bearing gap between the stated algorithmic procedure and the theoretical guarantee. The authors should either add an explicit Euler or other discretization analysis with controlled step-size, or clearly state that the theorem concerns the idealized exact-flow procedure and provide a separate discrete-time statement.
  2. [Theorem 2.2] Theorem 2.2 states σ ≍ n^{-4/(d+4)}, but the bandwidth used in Algorithm 1 and Theorem 2.5 is h = n^{-1/(d+4)}, and the proof of Theorem 2.5 uses σ' = h(1+O(h^2)). With σ = n^{-4/(d+4)}, the displayed bound (d(k+1) log n σ)^2 is O(n^{-8/(d+4)} log^2 n), which is not the rate used anywhere in the main argument and does not match the concentration claimed in Theorem 4.2. As printed, the formal 'projection' theorem is inconsistent with its application. This should be corrected to σ = n^{-1/(d+4)} (or the relation σ' = h(1+O(h^2)) should be made explicit in the statement).
  3. [Corollary 4.1.2 and event A0 in the proof of Theorem 2.5] Corollary 4.1.2 claims inf_{y∈M} |I_{y,σ}| ≥ 1 with probability 1-σ^k. For σ = n^{-1/(d+4)} this is only 1-n^{-k/(d+4)}, which is not the high-probability event 1-n^{-k} required to control A0 in the proof of Theorem 2.5. The net argument also does not appear to produce the claimed confidence after the union over the net. This is repairable by choosing a net whose cardinality and per-point failure probability combine to n^{-k}, but as written the event A0^c is not established at the level used.
  4. [Proof of Theorem 2.5, union bound over A_j] In the proof of Theorem 2.5, the events A_j = {∥F_{X,σ'}(X^{(j)}+σ'ξ) - F_{X,σ'}(exp_{X^{(j)}}(σ'ξ_T))∥ ≥ ...} each have probability at most n^{-k} by Theorem 4.2. The paper then writes P(A|X) ≤ ∑_j P(A_j|X) ≤ n^{-k}, but the union over n points gives n^{1-k}, not n^{-k}. This is a genuine union-bound error in the main proof. It can be fixed by applying Theorem 4.2 with failure probability n^{-(k+1)} per point and reindexing k, but the printed derivation overstates the confidence by a factor of n.
minor comments (5)
  1. [Section 2.2] The name 'Naradaya–Watson estimator' is misspelled; it should be 'Nadaraya–Watson estimator'.
  2. [Theorem 2.2] The variable r>0 in the statement of Theorem 2.2 is never used in the conclusion; it should be removed or explained.
  3. [Proof of Proposition 4.8] The phrase 'pick a σ(4+d)-net' is garbled; the intention appears to be a net of radius σ^{4+d} or similar. As printed, the notation is confusing.
  4. [Lemma 4.10 and Theorem 2.5] Theorem 2.5 states T > C(log(Dnk)) while Lemma 4.10 derives T = k(log n + log D + log(diam(M))). These should be aligned, with all constants and the role of k made consistent.
  5. [Section 4.4] In Lemma 4.5 and the proof of Proposition 2.3, the same symbol σ is used for the kernel bandwidth and for the noise level; this is standard but slightly overburdened, and the reader must track the distinction carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation chain is self-contained and anchored to external benchmarks.

full rationale

The paper's central claim, Theorem 2.5, is derived through a chain of auxiliary results that do not assume the target rate. The inertia update is first reduced to a Nadaraya–Watson estimator (Section 2.2), whose behavior near the manifold is analyzed in Theorem 4.2 using a Gaussian tail argument and a Pythagorean decomposition, not by assuming the final Wasserstein bound. The NWE is then compared to manifold KDE through Proposition 4.8, Lemma 4.9, and Proposition 2.3. The KDE L-infinity bound follows from standard bias–variance analysis (Lemmas 4.3 and 4.4) and Bernstein's inequality, and the W1-to-L-infinity step is attributed to a known optimal-transport inequality (Villani). The bandwidth h = n^{-1/(d+4)} is the standard minimax KDE bandwidth, not a parameter fitted to force the theorem. The empirical-measure rate W1(p̂0, p0) = O(n^{-1/d}) is quoted from Fournier–Guillin as an external benchmark. Self-citations ([14], [20]) appear only in the literature review and are not load-bearing for any proof step. The net-counting argument cites an external manifold-packing bound from Azangulov et al., not from the authors' prior work. The paper therefore does not rename a known result or smuggle an ansatz through self-citation: the inertial diffusion model is genuinely shown to approximate a manifold KDE, and the final rate is a consequence of that reduction. There are proof-technical issues flagged in the printed text, such as the union-bound exponent in Corollary 4.1.2 and the idealized exact-ODE integration in Algorithm 1, but these are correctness or implementation gaps, not circularity: they do not make the theorem equivalent to its inputs by construction. Overall, the derivation is self-contained and independent of the paper's own prior claims.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The derivation is self-contained modulo standard nonparametric statistics and Riemannian geometry. The bandwidth choice h = n^{-1/(d+4)} is the minimax KDE bandwidth, not a fitted constant. The experimental bandwidth constant C0 is a hand-chosen tuning parameter that does not enter the theorem. The main added idealization is exact ODE simulation.

free parameters (1)
  • C0 = 0.8
    Multiplicative constant in the experimental bandwidth sigma = C0 n^{-1/(d+4)}; chosen by hand in Section 3.1 and the paper notes it may be tuned per problem. It does not affect the theorem, which holds for any constant bandwidth.
assumptions (5)
  • domain assumption Manifold hypothesis: data lie on a compact C2 d-manifold embedded in R^D with local isometry and bounded geometry, and the density is C2 and bounded from both sides (Assumption 2.1).
    The projection, KDE, and net arguments in Theorem 2.2, Proposition 2.3 and Theorem 2.5 all require this structure; without it the stated rate is not established.
  • standard math Positive reach and unique normal projection for the tau-neighborhood of M, used in Proposition 2.1 and the normal bundle decomposition of Theorem 4.2.
    C2 compact embedded manifolds have positive reach; the paper invokes [22] for projection regularity. This is a standard geometric fact, not proven in the paper.
  • standard math Manifold net cardinality bound: a sigma-net of a compact d-manifold has size O(sigma^{-d}) and related entropy bounds as in Proposition 3 of [1].
    Used in Corollary 4.1.2 and the sup-norm KDE proofs for the union bound over nets; the paper imports the bound from [1].
  • standard math Villani's inequality W1 ≤ ||p - q||_L∞ for measures with bounded support (Proposition 7.10 of [38]).
    Used to convert the L∞ KDE sup-norm bound into the Wasserstein-1 bound in Section 2.3 and Theorem 2.5; cited from [38].
  • domain assumption Exact simulation of the OU flow: Z_{T-h^2} follows p̂_{h^2} exactly, with no discretization error in the ODE of Algorithm 1.
    The proof of Theorem 2.5 starts from Z_{T-h^2} ~ p̂_{h^2}; numerical integrations are not analyzed, so the guarantee is for the idealized continuous-time process.

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Pith. "Pith review of Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces." pith.science (2026). https://pith.science/paper/XX4OSIWS

@misc{pith2026250502508,
  author       = {Pith},
  title        = {Pith review of: Resolving Memorization in Empirical Diffusion Model for Manifold Data in High-Dimensional Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XX4OSIWS}},
  note         = {Machine review of arXiv:2505.02508}
}
abstract

Diffusion models are popular tools for generating new data samples, using a forward process that adds noise to data and a reverse process to denoise and produce samples. However, when the data distribution consists of n points, empirical diffusion models tend to reproduce existing data points, a phenomenon known as the memorization effect. Current literature often addresses this with complex machine learning techniques. This work shows that the memorization issue can be solved simply by applying an inertia update at the end of the empirical diffusion simulation. Our inertial diffusion model requires only the empirical score function and no additional training. We demonstrate that the distribution of samples from this model approximates the true data distribution on a $C^2$ manifold of dimension $d$, within a Wasserstein-1 distance of order $O(n^{-\frac{2}{d+4}})$. This bound significantly shrinks the Wasserstein distance between the population and empirical distributions, confirming that the inertial diffusion model produces new and diverse samples. Remarkably, this estimate is independent of the ambient space dimension, as no further training is needed. Our analysis shows that the inertial diffusion samples resemble Gaussian kernel density estimations on the manifold, revealing a novel connection between diffusion models and manifold learning.

Figures

Figures reproduced from arXiv: 2505.02508 by the authors.

Figure 1
Figure 1. Illustration of the inertia update. Training data (black boxes) are [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. W1 error of IDM vs memorized DM for data generated from the Haar measure on SO(4) embedded in R D. Both plots are in log-log scale. (Left) W1 loss w.r.t. sample size n with D = 50. Theoretical predictions for the algorithms are shown as dotted and dashed reference lines. (Right) W1 error w.r.t. embedding dimension D with fixed n = 2048. 3.1 Generation performance with different sample size We first validate that IDM… view at source ↗

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Works this paper leans on

41 extracted references · 18 canonical work pages

  1. [1]

    Conver- gence of diffusion models under the manifold hypothesis in high-dimensions

    Iskander Azangulov, George Deligiannidis, and Judith Rousseau. Conver- gence of diffusion models under the manifold hypothesis in high-dimensions. arXiv preprint arXiv:2409.18804, 2024

    arXiv 2024

  2. [2]

    Memorization and regularization in generative diffusion models.arXiv preprint arXiv:2501.15785, 2025

    Ricardo Baptista, Agnimitra Dasgupta, Nikola B Kovachki, Assad Oberai, and Andrew M Stuart. Memorization and regularization in generative diffusion models.arXiv preprint arXiv:2501.15785, 2025

    arXiv 2025

  3. [3]

    Laplacian eigenmaps for dimensionality reduction and data representation.Neural Computation, 15(6):1373–1396, 2003

    Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation.Neural Computation, 15(6):1373–1396, 2003

    2003

  4. [4]

    Cambridge University Press, 2023

    Nicolas Boumal.An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, 2023

    2023

  5. [5]

    Cambridge Studies in Advanced Mathematics

    Isaac Chavel.Riemannian Geometry: A Modern Introduction. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 2006

    work page 2006

  6. [6]

    Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.arXiv preprint arXiv:2209.11215, 2022

    Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru R Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.arXiv preprint arXiv:2209.11215, 2022

    arXiv 2022

  7. [7]

    Diffusion maps.Applied and computational harmonic analysis, 21(1):5–30, 2006

    Ronald R Coifman and Stéphane Lafon. Diffusion maps.Applied and computational harmonic analysis, 21(1):5–30, 2006

    2006

  8. [8]

    Diffusion models beat gans on imagesynthesis.Advances in neural information processing systems, 34:8780– 8794, 2021

    Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on imagesynthesis.Advances in neural information processing systems, 34:8780– 8794, 2021

    work page 2021

Show all 41 references
  1. [9]

    Minimax adaptive estimation in manifold inference.Elec- tronic Journal of Statistics, 15(2):5888–5932, 2021

    Vincent Divol. Minimax adaptive estimation in manifold inference.Elec- tronic Journal of Statistics, 15(2):5888–5932, 2021

    2021

  2. [10]

    Measure estimation on manifolds: an optimal transport approach.Probability Theory and Related Fields, 183(1):581–647, 2022

    Vincent Divol. Measure estimation on manifolds: an optimal transport approach.Probability Theory and Related Fields, 183(1):581–647, 2022

    2022

  3. [11]

    Testing the manifold hypothesis.Journal of the American Mathematical Society, 29(4):983–1049, 2016

    Charles Fefferman, Sanjoy Mitter, and Hariharan Narayanan. Testing the manifold hypothesis.Journal of the American Mathematical Society, 29(4):983–1049, 2016. 33

    2016

  4. [12]

    Interpolating between optimal transport and mmd using sinkhorn divergences

    Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-ichi Amari, Alain Trouve, and Gabriel Peyré. Interpolating between optimal transport and mmd using sinkhorn divergences. InThe 22nd International Conference on Artificial Intelligence and Statistics, pages 2681–2690, 2019

    2019

  5. [13]

    On the rate of convergence in wasser- stein distance of the empirical measure.Probability theory and related fields, 162(3):707–738, 2015

    Nicolas Fournier and Arnaud Guillin. On the rate of convergence in wasser- stein distance of the empirical measure.Probability theory and related fields, 162(3):707–738, 2015

    2015

  6. [14]

    Localized diffusion models for high dimensional distributions generation.arXiv preprint arXiv:2505.04417, 2025

    Georg A Gottwald, Shuigen Liu, Youssef Marzouk, Sebastian Reich, and Xin T Tong. Localized diffusion models for high dimensional distributions generation.arXiv preprint arXiv:2505.04417, 2025

    2025

  7. [15]

    Diffusion models as plug-and-play priors.Advances in Neural Information Processing Systems, 35:14715–14728, 2022

    Alexandros Graikos, Nikolay Malkin, Nebojsa Jojic, and Dimitris Samaras. Diffusion models as plug-and-play priors.Advances in Neural Information Processing Systems, 35:14715–14728, 2022

    2022

  8. [16]

    Kernel density estimation on riemannian manifolds: Asymptotic results.Journal of Mathematical Imaging and Vision, 34(3):235–239, 2009

    Guillermo Henry and Daniela Rodriguez. Kernel density estimation on riemannian manifolds: Asymptotic results.Journal of Mathematical Imaging and Vision, 34(3):235–239, 2009

    2009

  9. [17]

    Denoising diffusion probabilistic models.Advances in neural information processing systems, 33:6840–6851, 2020

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models.Advances in neural information processing systems, 33:6840–6851, 2020

    2020

  10. [18]

    Number 38

    Elton P Hsu.Stochastic analysis on manifolds. Number 38. American Mathematical Soc., 2002

    2002

  11. [19]

    An analytic theory of creativity in convolutional diffusion models.arXiv preprint arXiv:2412.20292, 2024

    Mason Kamb and Surya Ganguli. An analytic theory of creativity in convolutional diffusion models.arXiv preprint arXiv:2412.20292, 2024

    2024 arXiv

  12. [20]

    Temporal label recovery from noisy dynamical data.arXiv preprint arXiv:2406.13635, 2024

    Yuehaw Khoo, Xin T Tong, Wanjie Wang, and Yuguan Wang. Temporal label recovery from noisy dynamical data.arXiv preprint arXiv:2406.13635, 2024

    2024 arXiv

  13. [21]

    Geometric structures arising from kernel density estimation on riemannian manifolds.Journal of Multivariate Analysis, 114:112–126, 2013

    Yoon Tae Kim and Hyun Suk Park. Geometric structures arising from kernel density estimation on riemannian manifolds.Journal of Multivariate Analysis, 114:112–126, 2013

    2013

  14. [22]

    Existence, uniqueness and regularity of the projection onto differentiable manifolds.Annals of global analysis and geometry, 60(3):559–587, 2021

    Gunther Leobacher and Alexander Steinicke. Existence, uniqueness and regularity of the projection onto differentiable manifolds.Annals of global analysis and geometry, 60(3):559–587, 2021

    2021

  15. [23]

    A good score does not lead to a good generative model.arXiv preprint arXiv:2401.04856, 2024

    Sixu Li, Shi Chen, and Qin Li. A good score does not lead to a good generative model.arXiv preprint arXiv:2401.04856, 2024

    2024 arXiv

  16. [24]

    Mathematical analysis of singularities in the diffusion model under the submanifold assumption

    Yubin Lu, Zhongjian Wang, and Guillaume Bal. Mathematical analysis of singularities in the diffusion model under the submanifold assumption. arXiv preprint arXiv:2301.07882, 2023. 34

    2023 arXiv

  17. [25]

    How to generate random matrices from the classical compact groups.arXiv preprint math-ph/0609050, 2006

    Francesco Mezzadri. How to generate random matrices from the classical compact groups.arXiv preprint math-ph/0609050, 2006

    2006 arXiv

  18. [26]

    Minimax estimation of smooth densities in wasserstein distance.The Annals of Statistics, 50(3):1519–1540, 2022

    Jonathan Niles-Weed and Quentin Berthet. Minimax estimation of smooth densities in wasserstein distance.The Annals of Statistics, 50(3):1519–1540, 2022

    2022

  19. [27]

    Diffusion models are minimax optimal distribution estimators

    Kazusato Oko, Shunta Akiyama, and Taiji Suzuki. Diffusion models are minimax optimal distribution estimators. InInternational Conference on Machine Learning, pages 26517–26582. PMLR, 2023

    2023

  20. [28]

    Submanifold density estimation

    Arkadas Ozakin and Alexander Gray. Submanifold density estimation. Advances in neural information processing systems, 22, 2009

    2009

  21. [29]

    Kernel density estimation on riemannian manifolds.Statis- tics & Probability Letters, 73(3):297–304, 2005

    Bruno Pelletier. Kernel density estimation on riemannian manifolds.Statis- tics & Probability Letters, 73(3):297–304, 2005

    2005

  22. [30]

    Score-based generative models detect manifolds.Ad- vances in Neural Information Processing Systems, 35:35852–35865, 2022

    Jakiw Pidstrigach. Score-based generative models detect manifolds.Ad- vances in Neural Information Processing Systems, 35:35852–35865, 2022

    2022

  23. [31]

    A geometric framework for understanding memorization in generative models

    Brendan Leigh Ross, Hamidreza Kamkari, Tongzi Wu, Rasa Hosseinzadeh, Zhaoyan Liu, George Stein, Jesse C Cresswell, and Gabriel Loaiza-Ganem. A geometric framework for understanding memorization in generative models. International Conference on Machine Learning, 2024

    2024

  24. [32]

    Roweis and Lawrence K

    Sam T. Roweis and Lawrence K. Saul. Nonlinear dimensionality reduction by locally linear embedding.Science, 290(5500):2323–2326, 2000

    2000

  25. [33]

    Closed-form diffusion models.arXiv preprint arXiv:2310.12395, 2023

    Christopher Scarvelis, Haitz Sáez de Ocáriz Borde, and Justin Solomon. Closed-form diffusion models.arXiv preprint arXiv:2310.12395, 2023

    2023 arXiv

  26. [34]

    From graph to manifold laplacian: The convergence rate

    Amit Singer. From graph to manifold laplacian: The convergence rate. Applied and Computational Harmonic Analysis, 21(1):128–134, 2006

    2006

  27. [35]

    Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456, 2020

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456, 2020

    2011 arXiv

  28. [36]

    Adaptivity of diffusion models to manifold struc- tures

    Rong Tang and Yun Yang. Adaptivity of diffusion models to manifold struc- tures. InInternational Conference on Artificial Intelligence and Statistics, pages 1648–1656. PMLR, 2024

    2024

  29. [37]

    Nonparametric estima- tors.Introduction to Nonparametric Estimation, pages 1–76, 2009

    Alexandre B Tsybakov and Alexandre B Tsybakov. Nonparametric estima- tors.Introduction to Nonparametric Estimation, pages 1–76, 2009

    2009

  30. [38]

    American Mathematical Soc., 2021

    Cédric Villani.Topics in optimal transportation, volume 58. American Mathematical Soc., 2021

    2021

  31. [39]

    Sharp asymptotic and finite-sample rates of convergence of empirical measures in wasserstein distance.Bernoulli, 25(4A):2620–2648, 2019

    Jonathan Weed and Francis Bach. Sharp asymptotic and finite-sample rates of convergence of empirical measures in wasserstein distance.Bernoulli, 25(4A):2620–2648, 2019. 35

    2019

  32. [40]

    Wasserstein proximal operators describe score-based generative models and resolve memorization.arXiv preprint arXiv:2402.06162, 2024

    Benjamin J Zhang, Siting Liu, Wuchen Li, Markos A Katsoulakis, and Stan- ley J Osher. Wasserstein proximal operators describe score-based generative models and resolve memorization.arXiv preprint arXiv:2402.06162, 2024

    2024 arXiv

  33. [41]

    The emergence of reproducibility and generalizability in diffusion models.arXiv preprint arXiv:2310.05264, 2023

    Huijie Zhang, Jinfan Zhou, Yifu Lu, Minzhe Guo, Peng Wang, Liyue Shen, and Qing Qu. The emergence of reproducibility and generalizability in diffusion models.arXiv preprint arXiv:2310.05264, 2023. 36

    2023 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.