{"id":"6ad21707-aa2c-442f-829b-41e53a8ef42b","arxiv_id":"2505.21640","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"A projection-based symmetric-manifold diffusion model with spatially-varying covariance reaches near-Euclidean per-step training cost and polynomial sampling guarantees, with faster training and better sample quality on the torus, SO(n), and U(n).","lead":"Diffusion models on curved manifolds usually pay for expensive heat-kernel computations; this paper projects Euclidean Brownian motion onto the manifold with a spatially-varying covariance, cutting per-iteration training cost to one gradient evaluation and near-linear-in-d arithmetic. If it holds up, manifold diffusion becomes almost as cheap as Euclidean diffusion for tori, spheres, rotation groups, and unitary groups at dimensions above 1000.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Printed training objective omits the 1/2 reverse-drift term and changes the score factor, so Theorem 2.2's hypothesis \"minimize our training objective\" is not met by Algorithm 1.","rationale":"The reader's verdict (REJECT) is supported by the printed inconsistency between the reverse SDE derivation and the training objective. I focus on the objective mismatch rather than the Markovian projection assertion because it is directly checkable and independently invalidates the central claim. The paper defines Algorithm 1 via the objective in Eq. (4)/Line 1; Lemma 6.2's Step 3 uses Eq. (34) as the drift target. These disagree on the torus by the nonzero term (1/2)z. Since Theorem 2.2 assumes minimizers of the training objective, and Algorithm 2 uses f-hat as the drift of the generated SDE, the accuracy bound does not follow from the stated algorithm. This is not a disagreement with consensus or a missing verification; it is an internal algebraic inconsistency. I do not claim the method cannot be repaired: correcting Eq. (4) to include (1/2)∇φ^T Z_t (and fixing the factor 2) and reproving the Markovian-projection step could make the central claim viable. The empirical runtime comparisons are not affected by this theoretical concern, and code availability is a positive; but the theoretical core as printed is unsupported. Hence the verdict stays REJECT with moderate confidence.","tokens_in":42792,"tokens_out":15037,"duration_ms":167176,"concrete_test":"On M=T^1, set φ(z)=z mod 2π so ∇φ=1 and ∇²φ=0. For b=0 and τ=T-t, the correct Itô-projected reverse drift from Eq. (1) is 1/2 z + ∇log q_{τ|0}(z|0) = 1/2 z - z/(1-e^{-τ}), while the printed Eq. (4) target is ∇log q_{τ|0}(z|0) = -z/(1-e^{-τ}). Substitute z=1, τ=1: the correct value is 1/2 - 1/(1-e^{-1}) ≈ -1.082, and the printed value is -1/(1-e^{-1}) ≈ -1.582. Since these differ by the nonzero function (1/2)z, the minimizers f^* of the two objectives are different; therefore an ε-minimizer of the objective implemented in Algorithm 1 need not be an ε-minimizer of the drift objective used in the proof of Theorem 2.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The theorem's accuracy guarantee holds only for f-hat and g-hat that minimize the training objective for the reverse SDE. The derivation in Lemma 6.2 gives the Itô-projected drift target (Eq. 34) as (∇φ)^T(H_t + 2∇log q_{T-t|0}(H_t|b)) + 1/2 tr(∇²φ), following the reverse OU SDE dH_t = (1/2 H_t + 2∇log q_{T-t}(H_t))dt + dW_t in Eq. (19). The objective actually printed in Eq. (4) and implemented in Algorithm 1, Line 1, is (∇φ)^T (Z - ψ(b)e^{-(T-t)/2})/(e^{-(T-t)}-1) + 1/2 tr(∇²φ) = (∇φ)^T ∇log q_{T-t|0} + 1/2 tr(∇²φ). The term (1/2)∇φ^T H_t is missing, and the score coefficient differs. Unless (∇φ)^T H_t vanishes identically—which it does not on T^d or for the unitary/orthogonal projections—an ε-minimizer of the printed objective is not close to f^* from Lemma 6.2. Algorithm 2 then simulates an SDE with the wrong drift, and the TV guarantee ∥ν-π∥_TV ≤ O(ε d^9 log(d/ε)) does not follow. This is an internal inconsistency in the paper as written, independent of the unproven Markovian projection step; both need repair.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a diffusion generative model for symmetric Riemannian manifolds (torus, sphere, SO(n), U(n)) by projecting an Ornstein-Uhlenbeck process Z_t in R^d onto the manifold through a map φ, and simulating an associated reverse SDE. The training objective in Eq. (4) and Algorithm 1 is claimed to be derivable from Itô's lemma, and the sampling Algorithm 2 is claimed to inherit a polynomial-in-d total variation guarantee from an ε-minimizer of that objective (Theorem 2.2, Corollary 2.3). The main advertised contributions are O(1) gradient evaluations and O(d^{ω/2}) arithmetic per training iteration, with empirical demonstrations on tori, SO(n) and U(n). The theoretical core, however, contains an internal inconsistency between the printed training objective and the objective derived in the proof, as well as an unproven Markovian-projection step, so the main accuracy/runtime claims are not established as written.","tokens_in":43164,"tokens_out":7321,"duration_ms":84166,"significance":"If the central derivation were correct, the paper would make a meaningful contribution: it would reduce the per-iteration training cost of manifold diffusion models to nearly Euclidean levels and provide the first polynomial-in-d sampling guarantee for SO(n) and U(n). The paper also ships code and presents empirical comparisons with RSGM, TDM, and Euclidean baselines, which is a strength. However, the main theorem is gated by the correctness of the training objective and by the claim that the projected time-reversed process satisfies the SDE in Eq. (2); both gates fail in the current manuscript. The eigenvalue-gap verification of Assumption 2.1 also has a direction error that undermines the stated Lipschitz constants. These are load-bearing issues, not presentation defects.","major_comments":[{"comment":"The objective actually minimized by Algorithm 1 does not match the objective derived in Lemma 6.2. Eq. (4) and Algorithm 1, Line 1 train f against (∇φ)^T (Z_{T-t} − ψ(b)e^{−(T−t)/2})/(e^{−(T−t)}−1) + (1/2)tr(∇²φ), which equals (∇φ)^T ∇log q_{T−t|0}(Z_{T−t}|b) + (1/2)tr(∇²φ). Lemma 6.2's Eq. (34), however, targets (∇φ)^T( Z_{T−t} + 2∇log q_{T−t|0}(Z_{T−t}|b) ) + (1/2)tr(∇²φ), and Eq. (19) for the reverse OU process implies the drift should be (1/2)Z_{T−t} + 2∇log q. Thus the printed objective is missing the (1/2)(∇φ)^T Z_{T−t} drift term and has the wrong coefficient on the score term. An ε-minimizer of the objective in Eq. (4) is therefore not an ε-minimizer of the objective used in Theorem 2.2's hypothesis, so the theorem's accuracy guarantee does not follow for the algorithm as implemented.","section":"§3, Eq. (4); Algorithm 1, Line 1; §6.1, Eqs. (19), (31), (34)"},{"comment":"The derivation assumes that Y_t = φ(H_t) satisfies the closed SDE dY_t = E[ dY_t | Y_t ] with drift f⋆ and diffusion g⋆. For a non-injective projection φ, a deterministic function of a Markov process is not generally Markov, and the conditional expectation of the infinitesimal increment does not in general produce a Markovian SDE for the projected process. The paper neither cites nor proves a Markovian projection theorem. This is not a cosmetic issue: for U(n) and SO(n), the spectral projection depends on the evolving eigenvalue gaps, so the projected process φ(H_t) is not obviously Markovian by itself. Without this step, Eq. (2), Lemma 6.2, and hence Theorem 2.2 are unsubstantiated.","section":"§3, Eq. (2); §6.1, Eqs. (22)–(25)"},{"comment":"The proof of Lemma 6.4 uses the wrong direction of the eigenvalue-gap bound. It states ∥∇φ∥ ≤ Σ_i 1/(λ_{i+1}−λ_i) ≤ d^{1.5}√t α^{−1/3}, and justifies this by the assertion that λ_{i+1}−λ_i ≤ α^{1/3}(1/√d)√t. An upper bound on the gaps yields a lower bound on the reciprocals 1/(λ_{i+1}−λ_i), not an upper bound, so the displayed inequality for Σ 1/(gap) does not follow. The high-probability statement (40) controls the probability that a gap is small; extracting a lower bound on gaps from it would give reciprocals of the form poly(d)√t/α^c, not d^{1.5}√t α^{−1/3}. The claimed constants L1, L2, and therefore the exponents in Corollary 2.3, are not justified.","section":"§6.3, Lemma 6.4"},{"comment":"The proof of Lemma 6.6 expresses f⋆(U,t) as an integral of (∇φ)^T ∇log q_{T−t|0}(z) + (1/2)tr(∇²φ) over Λ, but the drift derived in Eq. (31) and Eq. (34) also contains the term (1/2)(∇φ)^T H_t (or (∇φ)^T H_t in Eq. (34)). This term is absent from the integrated formula, so the Lipschitz bound in Eq. (47) is proved for a different function than the actual reverse-drift f⋆. Consequently, the passage from Assumption 2.1 to the c-Lipschitz condition used in Lemma 6.3 and Lemma 6.7 is not established.","section":"§6.4, Lemma 6.6"}],"minor_comments":[{"comment":"Table 3 is labeled U(n) but reports d=1225 for n=50, which is n(n−1)/2, the dimension of SO(50), whereas dim(U(50))=2500. The dimension convention should be stated consistently for U(n) and SO(n).","section":"Table 3 and §5"},{"comment":"The displayed bound W2(Ŷ_t,Y_t) ≤ (ρ²(Ŷ_0,Y_0)+ε)e^{ct} omits the infimum term and constants present in Lemma 6.3; this is presumably an informal sketch, but the mismatch should be flagged so readers can reconcile the overview with the full proof.","section":"§4.1, Eq. (12) and Lemma 6.3"},{"comment":"The thresholds for 'small' eigenvalue gaps are inconsistent: the text uses O(1/√d), O(1/n^{10}), and O(1/n^8) at different points without explaining the relations among them or how they combine in the two-body analysis.","section":"§6.7"},{"comment":"The displayed assumption lists the condition ∥d/dU ∇φ(x)∥_{2→2} ≤ L1 twice and never states the analogous condition for d/dU ∇²φ, although the text later uses such a bound; this appears to be a typographical error.","section":"Assumption 2.1"}],"recommendation":"reject","confidential_remarks":"The paper leans heavily on the authors' own prior random matrix theory results [24,25] for the eigenvalue-gap bounds that verify Assumption 2.1; those are published independent results, so I do not see circularity, but the editor may want to check that the new eigenvalue-gap work here is not merely a repackaging. The central issue is that the printed algorithm provably does not minimize the objective that the proof analyzes, and the Markovian projection step is asserted rather than proved for the paper's flagship examples. These are not minor fixable gaps within the scope of a revision; they call the main theorem into question as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is genuinely worth a look. Instead of heat kernels, the paper runs an OU process in the ambient space and projects it onto symmetric manifolds with a spatially-varying covariance that accounts for curvature. Combined with the sublinear parametrization of the covariance term, this gives a real per-iteration speedup over RSGM, TDM, and SCRD, and the experiments back the runtime claims. The random-matrix analysis for average-case Lipschitzness is also a serious piece of work.\n\nThe problem is load-bearing. The training objective in Eq. (4) and Algorithm 1 does not match the reverse SDE derived in Lemma 6.2 / Eq. (34). The correct drift target has an extra (∇φ)^T H_t term and a factor 2 on the score; the printed objective contains only the projected score plus the trace term. On the torus these differ because ∇φ is the identity there, so an ε-minimizer of the printed objective is not close to the f* required for the TV guarantee. This is not a typo: Theorem 2.2 assumes the models minimize 'our training objective,' but the algorithm minimizes a different one, and the sampling simulation uses the wrong drift.\n\nThere is a second, deeper gap. The paper asserts that Y_t = φ(H_t) follows an SDE with drift equal to the conditional expectation, but for a non-invertible projection a function of a Markov process is not generally Markov. No Markovian projection theorem is cited or proved, and this is the gate through which all later guarantees pass. The SO(n) extension is also only a sketch; Section 6.7 gives a heuristic picture of 'bad intervals' and Haar measure rather than a proof of the stated TV bound. The reliance on the authors' own random-matrix results for eigenvalue gaps is not itself a flaw, since those are published and independent, but it does place weight on results that not all referees will know.\n\nBottom line: the idea is promising, the experiments are consistent with the method working in practice, and the runtime improvement is real. But the theory as written is internally inconsistent. The paper deserves a serious referee—the approach is novel and likely fixable—but it should not be accepted without a corrected derivation, a proof of the Markovian projection step, and a complete SO(n) argument.","headline":"Promising manifold-diffusion method, but the printed training objective contradicts the reverse SDE used in the sampling guarantee; the central theorem is unsupported as written.","tokens_in":43704,"tokens_out":4971,"would_cite":false,"duration_ms":49381,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58J65","60J60","60H10","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A projection-based diffusion framework trains on symmetric manifolds with one gradient evaluation per step and provably polynomial sampling error, closing most of the runtime gap with Euclidean diffusion.","keywords":["diffusion models","symmetric manifolds","Riemannian manifold diffusion","heat-kernel-free training","Brownian motion projection","average-case Lipschitzness","random matrix theory","optimal transport coupling"],"falsifier":"Take a simple non-invertible case, the sphere projection $\\varphi(z)=z/\\|z\\|$ in $d=3$: simulate the Ornstein-Uhlenbeck process $Z_t$ honestly, record the projected path $Y_t=\\varphi(Z_t)$, and estimate its short-time transition kernels; separately solve SDE (2) with the paper's conditional-expectation drift and covariance, and compare the transition kernels. A measurable discrepancy at small times would show that the SDE representation every accuracy bound flows from is not the true projected process.","tokens_in":2310,"feed_emoji":"🌀","tokens_out":2287,"duration_ms":111917,"temperature":0.7,"pith_summary":"This paper sets out to close the runtime gap between diffusion generative models in Euclidean space and those constrained to symmetric Riemannian manifolds: the torus, the sphere, the special orthogonal group $SO(n)$, and the unitary group $U(n)$. Existing manifold diffusion models lean on the heat kernel, which lacks a closed form and forces either exponential-in-dimension arithmetic or $d$ gradient evaluations per training step. The paper's construction chooses a forward diffusion with a spatially varying covariance so that the process is exactly the projection of Euclidean Ornstein-Uhlenbeck motion onto the manifold; training then minimizes a new objective, derived from Itô's Lemma, that costs one gradient evaluation and $O(d^{1.19})$ arithmetic per iteration on $SO(n)$ and $U(n)$. For trained models that $\\varepsilon$-minimize this objective, the proposed sampler is claimed to output a distribution within total-variation distance $O(\\varepsilon d^9 \\log(d/\\varepsilon))$ on $SO(n)/U(n)$, in $\\mathrm{poly}(d)\\cdot\\log(d/\\varepsilon)$ iterations. If these guarantees hold, manifold-constrained generation for robotics, drug discovery, and quantum-physics applications approaches the efficiency of Euclidean diffusion, with accuracy bounds that are polynomial rather than exponential in dimension.","feed_headline":"Diffusion on curved spaces trains at near-Euclidean speed","feed_subtitle":"On tori, spheres, and rotation/unitary groups, one gradient call per training step replaces d, with proven accuracy bounds.","key_machinery":"The load-bearing object is the projection map $\\varphi:\\mathbb{R}^d\\to M$ together with the Itô-Lemma projection of the Euclidean time-reversal: the reverse SDE $dY_t = f^*(Y_t,t)\\,dt + g^*(Y_t,t)\\,dB_t$ on $M$ has drift given by the conditional expectation of $\\nabla\\varphi(H_t)^\\top dH_t + \\tfrac12 dH_t^\\top \\nabla^2\\varphi(H_t) dH_t$ given $\\varphi(H_t)$, and covariance $g^*$ satisfying $(g^*)^2 = \\mathbb{E}[J\\varphi(H_t)^\\top J\\varphi(H_t) \\mid \\varphi(H_t)]$. Because the underlying Euclidean process has a closed-form Gaussian kernel, both terms are computable in closed form; for $SO(n)/U(n)$, $\\varphi$ is a singular value decomposition, and the covariance's symmetric structure reduces it to $n^2$ scalar functions $\\alpha_{ij} = \\mathbb{E}[1/(\\lambda_i-\\lambda_j)]$, which is what makes the per-iteration cost $O(d^{\\omega/2})$ instead of $d^2$. The proof machinery is an optimal-transport analysis: a Gronwall-type inequality for SDEs on non-negative curvature manifolds (Lemma 6.3), the random-matrix-theory verification of 'average-case Lipschitzness' (Assumption 2.1, Lemma 6.4), and a Wasserstein-to-total-variation conversion that resets the coupling every $\\tau = 1/c$ steps so error grows linearly rather than exponentially in time (Lemma 6.7).","core_discovery":"The paper's central claim is that the heat-kernel bottleneck of manifold diffusion can be replaced by a projection trick. The forward process is the projection $X_t = \\varphi(Z_t)$ of a Euclidean Ornstein-Uhlenbeck process $Z_t$ onto $M$, with a spatially varying covariance chosen so that this identity is exact even on curved manifolds, on which standard Brownian motion is not a projection. The reverse drift and covariance are then conditional expectations of the projected Euclidean reverse SDE, whose Gaussian transition kernel is closed-form; an indicator that excludes a high-probability 'average-case' set $\\Omega_t$ keeps the projection's derivative singularities (eigenvalue collisions, the sphere's origin) out of the picture. Manifold symmetries make $\\Omega_t$ depend only on the spectrum, so the conditional expectations smooth out the singularities and the true drift $f^*$ and covariance $g^*$ are $\\mathrm{poly}(d)$-Lipschitz on the whole manifold. On that basis, the paper claims Theorem 2.2: for $\\varepsilon$-minimizers $\\hat{f}$, $\\hat{g}$ of the training objective, Algorithm 2's output $\\nu$ satisfies $\\|\\nu-\\pi\\|_{TV} = O(\\varepsilon d^9 \\log(d/\\varepsilon))$ on $SO(n)/U(n)$, and $O(\\varepsilon d^6 \\log(d/\\varepsilon))$ on the torus and sphere, in $\\mathrm{poly}(d)\\cdot\\log(d/\\varepsilon)$ iterations, each needing one evaluation of $\\hat{f}$ and $\\hat{g}$, one exponential map, and $O(d)$ arithmetic operations.","pith_inferences":["The conditional-expectation (projected) SDE is the gate: a numerical check of whether $\\varphi(H_t)$'s transition kernels match SDE (2) on the sphere or a low-dimensional unitary group would directly stress-test the entire chain of guarantees.","The 'project plus average-case Lipschitz' template looks portable to other spaces admitting a projection oracle whose singular set is symmetry-invariant and rarely hit, such as the positive-semidefinite cone or shape spaces; the paper sketches polytopes but leaves this open.","A clean empirical test of the dimensional dependence is available: train with controlled objective values on wrapped Gaussians on tori of growing $d$, measure the TV error of the samples, and check whether it scales as $O(\\varepsilon d^6 \\log(d/\\varepsilon))$ as Theorem 2.2 predicts."],"forward_implications":["Per training step on $SO(n)$ and $U(n)$ drops to one gradient evaluation and $O(d^{1.19})$ arithmetic, versus $d$ gradient evaluations and exponential-in-$d$ heat-kernel sums in earlier manifold diffusion models.","Sampling accuracy becomes polynomial: with an $\\varepsilon$-optimal model the output is within $O(\\varepsilon d^9 \\log(d/\\varepsilon))$ total variation on $SO(n)/U(n)$ in $O(d^{5.5}\\log(d/\\varepsilon))$ iterations, each iteration requiring one drift/covariance evaluation plus one exponential map.","The forward diffusion is simulation-free: a single Gaussian draw plus one projection (an SVD for matrix groups) replaces numerical SDE/ODE solvers inside the training loop.","The framework composes under direct products, so it covers spaces like $SE(n) \\cong \\mathbb{R}^n \\times SO(n)$ used in rigid-body and protein-backbone generation.","Empirically, per-iteration training stays within a factor of 3 of a Euclidean diffusion model on $U(n)$ up to manifold dimension 1225, and C2ST and likelihood scores beat the prior heat-kernel and momentum-based baselines for $n \\geq 9$."],"supporting_citations":[{"why":"Supplies the prior Riemannian score-based diffusion baseline whose heat-kernel expansions (exponential-in-$d$) and non-polynomial sampling bounds are the bottleneck this paper removes.","marker":"[12]"},{"why":"Supplies the reverse-time SDE formula (Equation (1)) for identity-covariance diffusions that the paper projects onto the manifold via Itô's Lemma.","marker":"[1]"},{"why":"Supplies Proposition 20, the Lipschitz bound on the Euclidean score used in Lemma 6.6 to prove that $f^*$ and $g^*$ are $\\mathrm{poly}(d)$-Lipschitz everywhere.","marker":"[8]"},{"why":"Supplies the Ornstein-Uhlenbeck mixing bound used to set $T = \\log(dC/\\varepsilon)$ so that the initial distribution is within $\\varepsilon$ of the target.","marker":"[3]"},{"why":"Supplies the random-matrix eigengap bounds used to verify the average-case Lipschitz assumption (Assumption 2.1) on the unitary group.","marker":"[2]"},{"why":"Supplies eigenvalue-gap lower bounds for Gaussian perturbations used in Lemma 6.4 to control $\\nabla\\varphi$ and $\\nabla^2\\varphi$ on the high-probability set $\\Omega_t$.","marker":"[24]"},{"why":"Supplies Dyson Brownian motion, the eigenvalue and eigenvector SDEs used to extend the sampling proof to $SO(n)$ across near-collision time intervals.","marker":"[13]"},{"why":"Supplies the momentum-based Lie-group diffusion baseline whose implicit-score objective costs $d$ gradient evaluations per step and is compared against empirically.","marker":"[37]"}],"fun_headline_variants":["One gradient per step for diffusion on curved spaces","Projection bypasses heat kernels for faster manifold diffusion","Curved-space diffusion trains at near-Euclidean speed","Symmetric manifold diffusion without heat kernel computations","Projection trick makes manifold diffusion training O(1) per step"],"cache_read_input_tokens":45696,"weakest_assumption_plain":"The whole guarantee passes through the step where the projected time-reversed process $Y_t = \\varphi(H_t)$ is asserted to follow the SDE whose drift and covariance are those conditional expectations; when the projection is many-to-one, a deterministic function of a Markov process is not automatically Markov, and the paper does not cite or prove a theorem licensing that SDE representation.","fun_headline_variants_meta":{"raw":{"variants":["One gradient per step for diffusion on curved spaces","Projection bypasses heat kernels for faster manifold diffusion","Curved-space diffusion trains at near-Euclidean speed","Symmetric manifold diffusion without heat kernel computations","Projection trick makes manifold diffusion training O(1) per step"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000804,"raw_usage":{"total_tokens":3598,"prompt_tokens":1079,"completion_tokens":2519,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":695,"completion_tokens_details":{"reasoning_tokens":2442}},"tokens_in":695,"tokens_out":2519,"duration_ms":18978,"temperature":1.0,"reasoning_tokens":2442,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:31:22.068467+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a simple non-invertible case, the sphere projection $\\varphi(z)=z/\\|z\\|$ in $d=3$: simulate the Ornstein-Uhlenbeck process $Z_t$ honestly, record the projected path $Y_t=\\varphi(Z_t)$, and estimate its short-time transition kernels; separately solve SDE (2) with the paper's conditional-expectation drift and covariance, and compare the transition kernels. A measurable discrepancy at small times would show that the SDE representation every accuracy bound flows from is not the true projected process.","supporting_citations":[{"cited_title":"Riemannian score-based generative modelling.Advances in Neural In- formation Processing Systems, 35:2406–2422, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the prior Riemannian score-based diffusion baseline whose heat-kernel expansions (exponential-in-$d$) and non-polynomial sampling bounds are the bottleneck this paper removes."},{"cited_title":"Springer, 2014","cited_arxiv_id":null,"evidence_quote":"Supplies the Ornstein-Uhlenbeck mixing bound used to set $T = \\log(dC/\\varepsilon)$ so that the initial distribution is within $\\varepsilon$ of the target."},{"cited_title":"Number 118","cited_arxiv_id":null,"evidence_quote":"Supplies the random-matrix eigengap bounds used to verify the average-case Lipschitz assumption (Assumption 2.1) on the unitary group."},{"cited_title":"Private covariance approximation and eigenvalue- gap bounds for complex Gaussian perturbations","cited_arxiv_id":null,"evidence_quote":"Supplies eigenvalue-gap lower bounds for Gaussian perturbations used in Lemma 6.4 to control $\\nabla\\varphi$ and $\\nabla^2\\varphi$ on the high-probability set $\\Omega_t$."},{"cited_title":"A Brownian-motion model for the eigenvalues of a random matrix.Journal of Mathematical Physics, 3(6):1191–1198, 1962","cited_arxiv_id":null,"evidence_quote":"Supplies Dyson Brownian motion, the eigenvalue and eigenvector SDEs used to extend the sampling proof to $SO(n)$ across near-collision time intervals."},{"cited_title":"Triv- ialized momentum facilitates diffusion generative modeling on Lie groups","cited_arxiv_id":null,"evidence_quote":"Supplies the momentum-based Lie-group diffusion baseline whose implicit-score objective costs $d$ gradient evaluations per step and is compared against empirically."}],"review_version":1}