Generative model for optimal density estimation on unknown manifold

Arthur St\'ephanovitch

REVIEW 3 major objections 3 minor 20 references

Generative model for optimal density estimation on unknown manifold

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proposes a generative estimator that, on unknown smooth manifolds, simultaneously achieves the minimax-optimal rate for every Hölder IPM of order $\gamma \ge 1$.

desk verdict Ambitious and mostly coherent claim of simultaneous minimax optimality for Hölder IPMs on unknown manifolds, but the proof of the key n^{-1/2} rate has a real gap exactly in d=2, the dimension used in the experiments. read the letter →

arxiv 2506.19587 v1 pith:DX47LI3W submitted 2025-06-24 math.ST stat.TH

Arthur Stéphanovitch This is my paper · ORCID

classification math.STstat.TH MSC 62G0562G0762R30

keywords densityestimationunknownmanifoldsminimaxoptimalityHölderintegralprobabilitymetricsgenerativeadversarialnetworksgeometricWhitneyproblemwaveletapproximationmanifoldlearning

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

This paper claims that a single tractable, generative estimator can be minimax-optimal for density estimation on unknown low-dimensional manifolds, simultaneously for every Hölder integral probability metric of order $\gamma \ge 1$. Earlier estimators had to be retuned for one fixed $\gamma$, or were limited to tori, or were computationally intractable; the proposed construction instead reconstructs the support manifold at the same regularity as the truth and puts a smooth density on it. If correct, one training procedure automatically achieves the optimal rate $n^{-(\beta+\gamma)/(2\beta+d)} \vee n^{-1/2}$ (up to logarithms) for all $\gamma \ge 1$, which is what would make the result matter for practice and theory alike.

What carries the argument

The load-bearing objects are the local chart decomposition, the gluing map $F_{g,\varphi}$ built from a geometric reconstruction procedure, and the wavelet-truncated function classes $\mathcal{G}$, $\Phi$, $\mathcal{D}$ that parametrize generators, approximate inverses, and discriminators while keeping Hölder regularity under control. The critical discriminator smoothness is $d/2$, chosen so the adversarial loss drives the $d_{H^{d/2}_1}$ distance down to the parametric $n^{-1/2}$ rate; the interpolation inequality then converts the resulting closeness into optimal rates for all $\gamma \ge 1$, provided both the truth and the estimator satisfy the manifold and density regularity conditions. The density lower bound $f_\mu \ge K^{-1}$ is what triggers the regularity theory of optimal transport maps, which produces the diffeomorphic transport maps that absorb the target density into the Gaussian reference measure.

What would settle it

Construct a $\beta$-Hölder density on the 2-sphere that is zero on a small open cap and positive elsewhere, satisfying every other assumption; if the proposed estimator's expected $d_{H^\gamma_1}$ error no longer follows the claimed rate, or if its estimated density becomes unbounded or non-smooth near the zero patch, the optimal-transport regularity step is falsified. A second check uses a manifold whose reach is below the assumed threshold, such as two spheres joined by a sharp neck; the geometric gluing and the $n^{-1/2}$ rate for $d_{H^{d/2}_1}$ should fail.

Watch

Extended reading notes

Core claim

Under Assumption 1 — a $\beta$-Hölder density bounded below on a closed $(\beta+1)$-smooth $d$-dimensional submanifold of $\mathbb{R}^p$ — the paper constructs $\hat\mu = (F_{\hat g,\hat\varphi} \circ \hat g)_{\#\hat\alpha}\gamma_n^d$ by adversarially training wavelet-parametrized local charts and gluing them with the geometric reconstruction map $F_{g,\varphi}$. The main result (Theorem 5) states that with high probability the estimator satisfies Assumption 1 itself and that $\mathbb{E}[d_{H^\gamma_1}(\hat\mu,\mu_\star)] \le C \log(n)^{C_2}(n^{-(\beta+\gamma)/(2\beta+d)} \vee n^{-1/2})$ for all $\gamma \ge 1$, which is the known minimax rate up to logarithms. The mechanism is to reach the $n^{-1/2}$ rate at the critical smoothness $\gamma = d/2$ and then use an interpolation inequality for manifold-supported measures to transfer that rate to every $\gamma \ge 1$. A printed probability statement in Theorems 1 and 5 reads "at least $n^{-1}$"; the appendix proofs establish probability at least $1 - 1/n$.

Load-bearing premise

The load-bearing premise is that the target density never approaches zero on a closed manifold with at least $(\beta+1)$-smooth geometry and controlled reach; if the density vanishes somewhere or the manifold has a sharp crease, the smooth transport maps that carry a Gaussian into each chart, and hence the estimator's own smooth density, are not guaranteed.

Editorial extensions

If this is right

A single estimator matches the minimax rate for every Hölder IPM with $\gamma \ge 1$, removing the need to choose $\gamma$ before training.
The estimator is generative: sampling draws a truncated Gaussian latent vector, picks a chart, and applies the gluing map, so no stochastic differential equation solving is needed at inference.
The estimator inherits the same regularity as the target, giving a support manifold that is $(\beta+1)$-smooth and a density that is $\beta$-smooth and bounded below.
The wavelet parametrization makes the covering-number complexity depend on the intrinsic dimension $d$ rather than the ambient dimension $p$.
Integer values of $\beta$ cost only logarithmic factors in the rate and in the Hölder norm of the estimate.

Reading between the lines

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

If the lower bound on the density is relaxed to allow zeros, the optimal-transport regularity step is the likeliest point of collapse; a natural testable extension is to allow densities vanishing on small sets and see whether near-optimal rates survive with a slower constant.
The same geometric gluing mechanism could be applied to other reference measures and other critical metrics, suggesting that a single adversarially trained chart system might certify optimality for Wasserstein and maximum-mean-discrepancy distances as well.
The $n^{-1/2}$ rate at $\gamma = d/2$ is the engine of the simultaneous guarantee; if a different critical metric were used, the interpolation step would still lift the rate to all smoother IPMs, so the design principle is portable.
The empirical simplifications (directly learned charts, delayed gluing, smooth surrogate wavelets) are explicitly labelled as not covered by the theory; a reader could test whether the simplified model preserves rates on data with known manifold regularity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main claim is that one tractable generative estimator achieves minimax-optimal rates for every γ-Hölder IPM with γ ≥ 1 on unknown closed manifolds of arbitrary topology. That would be a genuine advance: Tang and Yang's estimator is infeasible, and the earlier Wasserstein GAN result was torus-only. The construction—Fefferman-style manifold gluing plus wavelet-parametrized local charts—is clever, and the high-level rate algebra is coherent: the discriminator smoothness d/2 creates a parametric pivot at n^{-1/2}, and interpolation is supposed to carry that to all γ ≥ 1. The lower bound is cited from Tang and Yang, not fitted, so there is no circularity. The paper deserves credit for assembling this machinery and for being honest about what is inherited from the author's own prior work.

The soft spot is load-bearing, and the stress-test note is right about it. The proof of Theorem 4 needs Proposition 9 to make the discriminator approximation gap decay. For d=2, γ=d/2=1, and the factor n^{-(d/2-1)/(2β+d)} becomes n^0, so the bound becomes E[d] ≤ C log^C (n^{-1/2} + E[d]). That is a tautology, and the subsequent bootstrap raises E[d] to the power (d/2-1)/(β+d/2), which is zero for d=2; the displayed conclusion does not follow. This is not a minor typo. The d>2 case appears to close, but the paper explicitly claims d≥2, and the numerical experiments are on the sphere and the torus—both 2-dimensional. So the unproven case is exactly the advertised practical regime.

Smaller issues: the printed "with probability at least n^{-1}" in Theorems 1 and 5 is vacuous; the proofs actually establish probability 1-1/n. Proposition 7's statement and proof do not quite match. The numerical section validates a simplified estimator, with no code and no error bars, which makes it suggestive rather than decisive. None of these alone would be fatal, but the d=2 gap is.

This paper is for mathematical statisticians working on manifold density estimation and GAN theory. If the d=2 case gets patched, it is likely a strong result. As it stands, the central claim is unproven in the dimension the authors advertise. I would still send it to peer review: the question is important, the machinery is substantial, and a serious referee is exactly what is needed to force the d=2 case to be fixed or the claim to be restricted. Desk rejection would be wrong.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes a generative estimator for a probability measure supported on an unknown d-dimensional submanifold of R^p. The estimator is built from multiple chart maps parameterized by low-frequency wavelets, glued by an operator inspired by Fefferman et al.'s geometric Whitney construction, and trained by an adversarial loss with a discriminator class of smoothness d/2. The main result (Theorem 5) claims that, for beta-regular densities on (beta+1)-smooth closed manifolds, the expected H^gamma_1-IPM error is bounded by polylog(n)(n^{-(beta+gamma)/(2beta+d)} vee n^{-1/2}) simultaneously for every gamma >= 1, which is the known minimax rate. The proof strategy is to obtain the n^{-1/2} rate for the pivot metric dH^{d/2}_1 (Theorem 4) and then interpolate to other gamma.

Significance. If the main theorem is fully proven, the paper is a substantial advance: it gives a single tractable, generative estimator that is simultaneously minimax-optimal for all gamma >= 1, removing the torus-topology restriction of Stephanovitch et al. (2024) and the fixed-gamma restriction of Tang and Yang (2023). The construction is ambitious and the appendix contains a detailed proof skeleton with careful attention to geometric regularity, wavelet parametrization, and empirical-process bounds; for d > 2 the rate algebra in the bootstrap of Theorem 4 is coherent. The use of Caffarelli regularity and the interpolation inequalities from the author's previous work is a plausible route. However, the proof of the central pivot result fails for d = 2, which is within the stated range and is the dimension of the paper's own experiments, so the contribution is not yet established as written.

major comments (3)

[B.3.2, proof of Theorem 4] The bootstrap controlling E[dH^{d/2}_1] collapses for d = 2. In the displayed chain after the bias-variance bound, the amplitude factor in front of E[dH^{d/2}_1^{(beta+1)/(beta+d/2)}] is n^{-(d/2-1)/(2beta+d)}; for d = 2 this is n^0 = 1. The recursion then reads E[dH^1_1] <= C log(n)^C (n^{-1/2} + E[dH^1_1]), which is a tautology, and the subsequent step raising the inequality to the power (d/2-1)/(beta+d/2) gives 1 <= C log(n)^C and yields no upper bound. Since Theorem 5 derives all gamma >= 1 rates by interpolating from dH^{d/2}_1 (Section B.3.3), the main result is unproven for d = 2.
[B.2.2, Proposition 9] The estimate ||nabla h_hat - nabla D_hat||_infty <= C ||h_hat - D_hat||_{B^{1,2}_{infty,infty}} is used to obtain the n^{-(d/2-1)/(2beta+d)} decay of the discriminator-approximation gap. This inequality is not valid for h_hat in H^1_1: the Besov norm B^{1,2}_{infty,infty} measures Zygmund-type smoothness of the function and does not control the L_infty norm of its gradient. A Lipschitz function such as h(x)=|x| has a discontinuous gradient, and its wavelet projection does not converge to the gradient in L_infty; hence the claimed decay is not a consequence of the stated assumptions. This is the root cause of the d = 2 failure in Theorem 4, and it is not a local typo.
[Section 3.2.3, Eq. (17)] The regularization term R(g) as defined is the count of pairs (z1,z2) that satisfy the desired near-isometry condition; the constraint R(g) <= epsilon_Gamma then favors maps that violate the condition. Lemma 6 and all subsequent uses require R(g) to count violations, i.e., R(g) = sum 1{... outside the allowed interval ...}. The displayed definition should be corrected to the complement before the geometric regularity results apply.

minor comments (3)

[Theorems 1 and 5; Proposition 9] The statements say 'with probability at least n^{-1}'; the proofs actually establish probability at least 1 - n^{-1}. Please correct the statements to match the proofs.
[Theorem 3] There is a typo: 'their exists' should be 'there exists'.
[Section 5] The simplifications (i)-(iv) are described as relaxing the theoretical guarantees; the paper should state explicitly that the experiments illustrate the practical heuristic version and do not validate the minimax theorem.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the minimax rate is derived from an internal bias-variance analysis plus an externally cited lower bound; self-citations are load-bearing but parameter-free prior theorems. A separate d=2 proof gap is a correctness issue, not a circular construction.

full rationale

The paper's central upper bound is not an input. The estimator is defined through the wavelet-parametrized classes (26)-(28), and the rate n^{-(β+γ)/(2β+d)} ∨ n^{-1/2} is derived in Theorem 3 from a bias-variance decomposition, with approximation errors controlled in Propositions 8 and 9 and complexity terms controlled by Lemma 3. The lower bound is cited externally to Tang and Yang (2023), so the minimax claim has independent grounding. The sentence in Section 2.3 that the estimator is 'specifically designed to achieve the minimax rate of n^{-1/2} for the metric dH^{d/2}_1' describes the design goal; it does not inject the rate as an assumption, and in the proof the n^{-1/2} term emerges from the bootstrap, not from the construction. The paper's many self-citations (Theorem 2 interpolation inequality, Corollary 1 Caffarelli transport map, Corollary 20 Wasserstein-to-IPM comparison, Proposition 3 wavelet embedding, Lemma 3 covering bounds) are indeed load-bearing, but each is a parameter-free theorem stated for general measures satisfying the manifold/density regularity conditions, without the target minimax rate as an assumption. Under the review rules, such citations count as independent support even when authored by the same researcher, so they do not create circularity. One non-circular caveat should be flagged: in the proof of Theorem 4 (Section B.3.2), for d=2 the exponent (d/2-1)/(2β+d) is zero, so the displayed bound becomes a tautological inequality E[dH^1_1] ≤ C log(n)^C (n^{-1/2} + E[dH^1_1]) and the subsequent division by (d/2-1) is division by zero. This leaves the claimed d≥2 result unproven for the two-dimensional case, which also covers the experiments; however, this is a mathematical proof gap rather than a circular reduction, because the estimator and the rate are not defined in terms of each other. Accordingly, the circularity score is low.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The construction's truth rests on the statistical model (Assumption 1), standard function-space facts (wavelet bases, Besov-Hölder embeddings, Stein extension), and several imported results from the author's own earlier work (interpolation inequality, transport-map existence, IPM separation lower bound). No free parameters are fitted to data in the theory; the scalings δn, δN and the sample allocation N are design choices made to balance the bias-variance decomposition, and they require β and d as known inputs. The experiments introduce additional hand-chosen hyperparameters (number of charts, network widths, wavelet dictionary sizes) that are not part of the theoretical claim. No new physical or statistical entities are postulated; the gluing operators, consistency loss, and regularization are algorithmic constructions defined in the paper, not entities with independent evidence requirements.

free parameters (3)

Smoothness β and intrinsic dimension d of the target class = assumed known (not estimated)
The wavelet truncation scales δn = n^{-1/(2β+d)} and δN = N^{-1/(2β+d)}, the discriminator regularity d/2, and the sample allocation N all require β and d as inputs. The guarantee is minimax over the known-smoothness class Pβ, not an adaptive bound.
Sample-size allocation N for fake samples = N = n if β + 1 ≥ d/2, else N = n^{(2β+d)/(4β+2)}
N is chosen in Section 4.2.1 to balance the variance of the composite class D∘G against the generator approximation error δN. It depends on β and d and is a design choice, not fitted to data.
Gluing cutoff εΓ and chart radius τ = τ = 1/(8K); εΓ ∈ (C1^{-2}, C1^{-1})
εΓ is the support scale of the smooth cutoff Γ in the gluing operators (12)-(14) and must lie in an interval fixed by the constants of the geometric lemmas (Propositions 6 and 7); τ is fixed by the regularity constant K in Assumption 1. These are structural constants, not data-fitted.

assumptions (7)

domain assumption Caffarelli regularity theory guarantees a Hölder-(β+1) diffeomorphism g2_i with λmin(∇g2_i) ≥ C^{-1} transporting the standard Gaussian to each chart pullback density (via Corollary 1 of Stéphanovitch (2024)).
Used in Proposition 1 (Section A.2.1) to construct the density maps; requires the target density bounded below by K^{-1} (Assumption 1 ii). If the density vanishes on part of the manifold, the smooth transport map is not guaranteed.
ad hoc to paper The interpolation inequality for Hölder IPMs on submanifolds (Theorem 2), quoted from Stéphanovitch (2024, arXiv:2406.01268), is correct.
Load-bearing for extending the n^{-1/2} rate at γ = d/2 to all γ ≥ 1 (Theorem 5, Section 4.2.3). It is not proved in this paper, not machine-checked, and its source is the same author's preprint.
ad hoc to paper Corollary 20 of Stéphanovitch (2024): supports separated by distance r imply d_{H^{d/2}_1} lower bounded by C^{-1} exp(-C r^{-2}).
Used in Proposition 11 and Lemma 9 to convert Wasserstein support-separation lower bounds into IPM lower bounds. Imported from the author's preprint without proof.
domain assumption Assumption 1: the target is a β-regular density (β > 1, d ≥ 2) bounded below, supported on a closed (β+1,K)-manifold in R^p.
Statistical model, Definitions 1 and 2. The density lower bound is crucial for Caffarelli regularity (stated in Section 2.2); d ≥ 2 is required by the construction.
standard math There exist compactly supported scaling and wavelet functions in H^{d/2+2}(R,R) forming an L2-orthonormal system (Daubechies wavelets).
Section 3.3.1; standard existence result (Daubechies 1988) with the required smoothness for the embedding arguments.
standard math Besov-Hölder embeddings: H^η = B^η_{∞,∞} for non-integer η, and the compact embeddings for integer η (Lemma 1).
Used throughout Section 3.3 to relate Besov and Hölder regularity; cited to Giné and Nickl (2015) and Haroske (2006).
standard math Stein's extension theorem (Theorem 4, chapter 6 of Stein 1970) extends local charts to maps in H^{β+1}(R^d, R^p).
Used in the proofs of Lemma 2 and Proposition 8 to extend local coordinate maps to global Hölder maps; standard.

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Pith. "Pith review of Generative model for optimal density estimation on unknown manifold." pith.science (2026). https://pith.science/paper/DX47LI3W

@misc{pith2026250619587,
  author       = {Pith},
  title        = {Pith review of: Generative model for optimal density estimation on unknown manifold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DX47LI3W}},
  note         = {Machine review of arXiv:2506.19587}
}

abstract

We propose a generative model that achieves minimax-optimal convergence rates for estimating probability distributions supported on unknown low-dimensional manifolds. Building on Fefferman's solution to the geometric Whitney problem, our estimator is itself supported on a submanifold that matches the regularity of the data's support. This geometric adaptation enables the estimator to be simultaneously minimax-optimal for all $ \gamma $-H\"older Integral Probability Metrics (IPMs) with $ \gamma \geq 1 $. We validate our approach through experiments on synthetic and real datasets, demonstrating competitive or superior performance compared to Wasserstein GAN and score-based generative models.

Figures

Figures reproduced from arXiv: 2506.19587 by the authors.

**Figure 2.** Comparison of the estimators for the uniform measure on the torus. The data are represented in dark [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Sample generated by our estimator after being trained on the MNIST dataset [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Review history (2 revisions) →

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Reference graph

Works this paper leans on

20 extracted references · 12 canonical work pages

[2]

Decomposition through charts

and bothfϕxi#µ⋆ andPm j=1χj(g1 j (·)) are bounded above and below onBd(0,τ ). Then, taking ξi = Ψ# ¯ζi, we have that its probability density is equal to ξi(x) = γd(x) exp(zi(Ψ−1(x))α−1 i , so from Corollary 1 in Stéphanovitch (2024), there exists a mapg2 i∈H β+1 C (Rd, Rd) such that (g2 i )#γd =ξi and λmin(∇g2 i )≥C−1. Therefore, we have Eµ⋆[h(X)] = mX i=...

work page 2024
[3]

Let w ∈M∩ Bp(x, (4C3)−1ϵΓ), l∈{ 1,...,m} and v∈Bd(0,τ ) such thatw =Fg,φ◦g1 l (v)

Let us show that Fg,φ m ◦...◦Fg,φ j+1◦gj is a Cβ+1 diffeomorphism between an open subset of Rd andM∩ Bp(x, (4C3)−1ϵΓ). Let w ∈M∩ Bp(x, (4C3)−1ϵΓ), l∈{ 1,...,m} and v∈Bd(0,τ ) such thatw =Fg,φ◦g1 l (v). From (49) we have ∥Fg,φ◦g1 l (v)−g1 j (0)∥≤∥ Fg,φ◦g1 l (v)−x∥ +∥x−g1 j (0)∥ ≤τ(1 + (K + 1)τ) + ϵΓ 2∥φj∥H1 and from Lemma 5 ∥Fg,φ j−1◦...◦Fg,φ 1 ◦g1 l (v)−g...

work page 2024
[6]

In the case whereΨ(u) /∈g2 j (Bd(0, log(n)1/2 + 1)), we haveζg j (u) = 0 so let us focus on the case whereΨ(u) belongs tog2 j (Bd(0, log(n)1/2 + 1))

Let us now show the upper bound onζg j. In the case whereΨ(u) /∈g2 j (Bd(0, log(n)1/2 + 1)), we haveζg j (u) = 0 so let us focus on the case whereΨ(u) belongs tog2 j (Bd(0, log(n)1/2 + 1)). Recalling that Ψ(u) := u τ 2−∥u∥2, we have ζj(u) =| det ∇((g2 j )−1◦ Ψ)(u) |γn d (g2 j )−1◦ Ψ)(u) ≤C 1 (τ 2−∥u∥2)2dγn d (g2 j )−1( u τ 2−∥u∥2 ) . (39) Now, asg2 j∈ ˆFd...

work page 2024
[7]

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

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,

arXiv 2011
[9]

Rong Tang and Yun Yang

URL https://arxiv.org/abs/2406.01268. Rong Tang and Yun Yang. Minimax rate of distribution estimation on unknown submanifolds under adversarial losses. The Annals of Statistics, 51(3):1282–1308,

arXiv
[12]

Let µ a probability measure satisfying Assumption 1 on a manifoldM. Then, there exists a collection of maps (ϕi)i=1,...,m∈H β+1 C (Bd(0,τ ), Rp)m satisfying∀u,v ∈ Bd(0,τ ), i∈{ 1,...,m}, 1−Kτ ≤ ∥∇ϕi(u) v ∥v∥∥ ≤1 + Kτ and there exists non-negative functions (ζi)i=1,...,m ∈ Hβ C(Rd, R)m with supp(ζi) ⊂ Bd(0,τ ), such that for all bounded continuous function...

work page 2024
[13]

sample of lawµ

Let n,A ∈ N>0, M satisfying the (β + 1,K )-manifold condition, µ a probability measure satisfying the (β,K )-density condition onM and (X1,...,X n) an i.i.d. sample of lawµ. Then for allγ≥ 1 and D⊂H γ A(Rp, R), we have E h sup D∈D EX∼µ[D(X)]− 1 n nX j=1 D(Xj) i ≤C min δ∈[0,1] ( A r (δ + 1/n)2 log(n|(D◦ ϕ)1/n|) n + AC2 √n 1 +δ(1− d 2(γ∧(β+1)) ) + log(δ−1)1...

work page 2023
[14]

LetG and Φ the classes of functions defined in(26) and (27) respectively. Then, for indepen- dent random variablesU1,...,U N∼U (Bd(0,τ )), T1,...,T N∼U m, we have E h sup (g,φ)∈G×Φ C(g,φ )−CN(g,φ ) i ≤C log(N)C2 min δ∈[0,1] (r (δ1 + 1/N)2 log(N|G1/N||(Φ◦G )1/N|) N + 1√ N (1 +δ(1− d 2(β+1) ) + log(δ−1)1{2(β+1)=d}) ) . Proof. For (g,φ )∈G× Φ and independent...

work page 2023

Show all 20 references

[15]

LetG, Φ,A,D be defined in(26), (27), (10) and (28) respectively. Then, for iid independent random variablesY1,...,Y N∼γn d, ω1,...,ω N∼U m, X1,...,X n∼µ⋆, we have E h sup (g,φ,α)∈G×Φ×A,D∈D L(g,φ,α,D )−LN,n(g,φ,α,D ) i ≤C log(N)C2 min δ1[0,1] (r (δ1 + 1/N)2 log(N|(D◦F G,Φ◦G )1/...

2023
[18]

Let πi be the projection on the tangent space ofM⋆ at the point g1 i (0) and Φ :πi◦g1 i (Bd(0, 2τ))→Bd(0, 2τ) defined by Φ(u) = (πi◦g1 i )−1(u)

In particular, we have thatg1 i = ϕ−1 g1 i (0) the inverse of the orthogonal projection on the tangent spaceTg1 i (0) ofM⋆ defined in (2). Let πi be the projection on the tangent space ofM⋆ at the point g1 i (0) and Φ :πi◦g1 i (Bd(0, 2τ))→Bd(0, 2τ) defined by Φ(u) = (πi◦g1 i )...

1970
[19]

Let πi be the projection on the tangent space ofM⋆ at the point g1 i (0) and Φi :πi◦g1 i (Bd(0, 2τ))→Bd(0, 2τ) defined by Φi(x) = (πi◦g1 i )−1(x)

In particular, we have thatg1 i =ϕ−1 g1 i (0) is the inverse of the orthogonal projection on the tangent spaceTg1 i (0) ofM⋆ defined in (2). Let πi be the projection on the tangent space ofM⋆ at the point g1 i (0) and Φi :πi◦g1 i (Bd(0, 2τ))→Bd(0, 2τ) defined by Φi(x) = (πi◦g1...

1970
[20]

Supposing thatµ⋆ satisfies Assumptions 1, in the caseN <A, we have dHd/2 1 (ˆµ,µ⋆)≤ 2≤ 2AN−1, (59) so we have the result

Proof of lemma 13.LetA> 0 be the constant given by Lemma 2 such that ifN≥A, there exists(g,φ )∈G× Φ satisfying the constraint (18). Supposing thatµ⋆ satisfies Assumptions 1, in the caseN <A, we have dHd/2 1 (ˆµ,µ⋆)≤ 2≤ 2AN−1, (59) so we have the result. Let us then focus on th...

2024
[1970]

URLhttp://www.jstor.org/stable/j.ctt1bpmb07

ISBN 9780691080796. URLhttp://www.jstor.org/stable/j.ctt1bpmb07. Arthur Stéphanovitch. Smooth transport map via diffusion process.arXiv preprint arXiv:2411.10235,

arXiv
[1997]

On the statistical properties of generative adversarial models for low intrinsic data dimension.arXiv preprint arXiv:2401.15801,

Saptarshi Chakraborty and Peter L Bartlett. On the statistical properties of generative adversarial models for low intrinsic data dimension.arXiv preprint arXiv:2401.15801,

arXiv
[2001]

A good score does not lead to a good generative model

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

arXiv
[2003]

Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh

doi: 10.1007/s00222-002-0255-6. Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh. Delaunay triangulation of manifolds.Foundations of Computational Mathematics, 18:399–431,

doi
[2006]

Alias-free generative adversarial networks.CoRR, abs/2106.12423,

Tero Karras, Miika Aittala, Samuli Laine, Erik Härkönen, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Alias-free generative adversarial networks.CoRR, abs/2106.12423,

arXiv
[2016]

Application of conditional ddpm on the mnist dataset

Xin Wang. Application of conditional ddpm on the mnist dataset. InFifth International Conference on Signal Processing and Computer Science (SPCS 2024), volume 13442, pages 124–128. SPIE,

2024
[2017]

Convergence of diffusion models under the manifold hypothesis in high-dimensions.arXiv preprint arXiv:2409.18804,

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

arXiv
[2024]

Score approximation, estimation and distribution recovery of diffusion models on low-dimensional data.arXiv preprint arXiv:2302.07194,

Minshuo Chen, Kaixuan Huang, Tuo Zhao, and Mengdi Wang. Score approximation, estimation and distribution recovery of diffusion models on low-dimensional data.arXiv preprint arXiv:2302.07194,

arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [2]

Decomposition through charts

and bothfϕxi#µ⋆ andPm j=1χj(g1 j (·)) are bounded above and below onBd(0,τ ). Then, taking ξi = Ψ# ¯ζi, we have that its probability density is equal to ξi(x) = γd(x) exp(zi(Ψ−1(x))α−1 i , so from Corollary 1 in Stéphanovitch (2024), there exists a mapg2 i∈H β+1 C (Rd, Rd) such that (g2 i )#γd =ξi and λmin(∇g2 i )≥C−1. Therefore, we have Eµ⋆[h(X)] = mX i=...

work page 2024

[2] [3]

Let w ∈M∩ Bp(x, (4C3)−1ϵΓ), l∈{ 1,...,m} and v∈Bd(0,τ ) such thatw =Fg,φ◦g1 l (v)

Let us show that Fg,φ m ◦...◦Fg,φ j+1◦gj is a Cβ+1 diffeomorphism between an open subset of Rd andM∩ Bp(x, (4C3)−1ϵΓ). Let w ∈M∩ Bp(x, (4C3)−1ϵΓ), l∈{ 1,...,m} and v∈Bd(0,τ ) such thatw =Fg,φ◦g1 l (v). From (49) we have ∥Fg,φ◦g1 l (v)−g1 j (0)∥≤∥ Fg,φ◦g1 l (v)−x∥ +∥x−g1 j (0)∥ ≤τ(1 + (K + 1)τ) + ϵΓ 2∥φj∥H1 and from Lemma 5 ∥Fg,φ j−1◦...◦Fg,φ 1 ◦g1 l (v)−g...

work page 2024

[3] [6]

In the case whereΨ(u) /∈g2 j (Bd(0, log(n)1/2 + 1)), we haveζg j (u) = 0 so let us focus on the case whereΨ(u) belongs tog2 j (Bd(0, log(n)1/2 + 1))

Let us now show the upper bound onζg j. In the case whereΨ(u) /∈g2 j (Bd(0, log(n)1/2 + 1)), we haveζg j (u) = 0 so let us focus on the case whereΨ(u) belongs tog2 j (Bd(0, log(n)1/2 + 1)). Recalling that Ψ(u) := u τ 2−∥u∥2, we have ζj(u) =| det ∇((g2 j )−1◦ Ψ)(u) |γn d (g2 j )−1◦ Ψ)(u) ≤C 1 (τ 2−∥u∥2)2dγn d (g2 j )−1( u τ 2−∥u∥2 ) . (39) Now, asg2 j∈ ˆFd...

work page 2024

[4] [7]

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

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,

arXiv 2011

[5] [9]

Rong Tang and Yun Yang

URL https://arxiv.org/abs/2406.01268. Rong Tang and Yun Yang. Minimax rate of distribution estimation on unknown submanifolds under adversarial losses. The Annals of Statistics, 51(3):1282–1308,

arXiv

[6] [12]

Let µ a probability measure satisfying Assumption 1 on a manifoldM. Then, there exists a collection of maps (ϕi)i=1,...,m∈H β+1 C (Bd(0,τ ), Rp)m satisfying∀u,v ∈ Bd(0,τ ), i∈{ 1,...,m}, 1−Kτ ≤ ∥∇ϕi(u) v ∥v∥∥ ≤1 + Kτ and there exists non-negative functions (ζi)i=1,...,m ∈ Hβ C(Rd, R)m with supp(ζi) ⊂ Bd(0,τ ), such that for all bounded continuous function...

work page 2024

[7] [13]

sample of lawµ

Let n,A ∈ N>0, M satisfying the (β + 1,K )-manifold condition, µ a probability measure satisfying the (β,K )-density condition onM and (X1,...,X n) an i.i.d. sample of lawµ. Then for allγ≥ 1 and D⊂H γ A(Rp, R), we have E h sup D∈D EX∼µ[D(X)]− 1 n nX j=1 D(Xj) i ≤C min δ∈[0,1] ( A r (δ + 1/n)2 log(n|(D◦ ϕ)1/n|) n + AC2 √n 1 +δ(1− d 2(γ∧(β+1)) ) + log(δ−1)1...

work page 2023

[8] [14]

LetG and Φ the classes of functions defined in(26) and (27) respectively. Then, for indepen- dent random variablesU1,...,U N∼U (Bd(0,τ )), T1,...,T N∼U m, we have E h sup (g,φ)∈G×Φ C(g,φ )−CN(g,φ ) i ≤C log(N)C2 min δ∈[0,1] (r (δ1 + 1/N)2 log(N|G1/N||(Φ◦G )1/N|) N + 1√ N (1 +δ(1− d 2(β+1) ) + log(δ−1)1{2(β+1)=d}) ) . Proof. For (g,φ )∈G× Φ and independent...

work page 2023