{"id":"d70c374a-7e0c-404a-9251-6d500f698f64","arxiv_id":"2506.18290","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Diffusion reconstruction errors are explained by an instability of the probability-flow ODE: under sparsity assumptions, the chance of instability tends to one as image dimension grows.","lead":"This paper argues that diffusion reconstruction errors are amplified by an intrinsic instability of the generation mapping, caused by the sparsity of the generated image distribution. It formalizes the instability and proves that, under stated assumptions, it becomes almost certain in high dimensions, with experiments on Stable Diffusion and FLUX.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of δ→0 in Theorem 4.3 replaces π_real by π_gen in the tail estimate; without a uniform bound on p_real/p_gen, the conclusion that inverted real noise is Gaussian-concentrated is exactly what remains unproven.","rationale":"I read the paper as attempting to show that reconstruction errors in diffusion models are amplified by an intrinsic instability of the PF-ODE, and that this instability is almost certain in high dimension (Theorem 4.3). The reader's verdict is REJECT, and my stress test points to the same load-bearing weakness: the δ→0 part of Theorem 4.3 is not established. The intuition that Gaussian noise concentrates near a sphere is applied to G^{-1}(x) for x ~ π_real, but no assumption connects π_real to π_gen strongly enough to transfer Gaussian concentration to the inverse map. The appendix's invocation of absolute continuity is insufficient because the Radon–Nikodym derivative p_real/p_gen may be unbounded, and precisely in the low-density regions where p_gen is exponentially small the ratio can explode. Writing δ_n as an integral over z makes this explicit: the integrand contains p_real(G(z))/p_gen(G(z)), and controlling it requires a regularity or closeness condition that is not among Assumptions 4.1–4.4. I therefore do not see the central theorem as proved. This is an internal gap rather than merely a disagreement with the realism of Assumption 4.2; even accepting the quoted assumptions, the proof does not go through. The empirical correlation evidence is suggestive but secondary. I would keep the reader's REJECT verdict unchanged.","tokens_in":22240,"tokens_out":9359,"duration_ms":100285,"concrete_test":"Independently re-derive the δ→0 step in Appendix D.5 starting from δ_n = ∫_{||z||² > 2n+3√(2n)} γ(z) [p_real(G(z))/p_gen(G(z))] dz and using only Assumptions 4.1–4.4, attempting to obtain an explicit sequence of upper bounds that tends to zero. To test the bound numerically, implement the flow-matching ODE for a target π_gen given by a mixture of Gaussian neighbors (centers inside [0,1]^n, widths w_i ≪ 1) with π_real uniform on [0,1]^n, for n = 16, 32, 64, and estimate δ_n by Monte Carlo over real samples. If δ_n does not decrease toward zero, Theorem 4.3 is false as stated; if it does decrease, the calculation will reveal which suppressed bound on p_real/p_gen makes it work.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central asymptotic claim is Theorem 4.3, and its proof splits into ε→0 and δ→0. The δ step is not derived from the stated assumptions. δ_n := π_real({x : ||G^{-1}(x)||^2 > 2n+3√(2n)}) is a statement about π_real, but Appendix D.5 computes a tail probability under π_gen ('Since π_real(n) is absolutely continuous about π_gen ... δ′ = π_gen(...)') and then transfers it back to π_real via a Kolmogorov-extension limit. Absolute continuity alone gives no quantitative control. With x = G(z), the change of variables yields δ_n = ∫_{||z||^2 > 2n+3√(2n)} γ(z) [p_real(G(z))/p_gen(G(z))] dz. Under Assumption 4.2, p_real is bounded above on [0,1]^n, but under Assumptions 4.3–4.4, p_gen is a sparse mixture of Gaussian neighbors whose density can be exponentially small on most of the cube, so the density ratio p_real/p_gen need not be bounded uniformly in n. The desired conclusion δ_n→0 is therefore equivalent to a strong closeness condition between π_real and π_gen—that the preimage under the generation map of a uniform real distribution lies asymptotically in the Gaussian concentration ball. That condition is not a consequence of Assumptions 4.1–4.4. Since P_M ≥ 1 − ε − δ, failure of δ→0 leaves the theorem without its conclusion. Replacing π_real with π_gen at this step is the load-bearing move, and it is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper argues that errors in diffusion-based image reconstruction are amplified by an intrinsic instability of the PF-ODE generation map, caused by the sparsity of the generation distribution. It defines an intrinsic instability coefficient and a geometric-average metric, provides toy and Stable Diffusion/FLUX experiments showing a positive correlation between instability and reconstruction error, and claims a theorem (Theorem 4.3) that, under assumptions on the real and generated distributions, the probability of instability tends to one as the data dimensionality grows. The main theoretical contribution is the asymptotic analysis in Section 4, with the proofs in Appendix D.","tokens_in":22532,"tokens_out":4776,"duration_ms":43731,"significance":"If Theorem 4.3 were correct, the result would be significant: it would provide a rigorous mechanism by which ideal diffusion reconstruction of high-dimensional images is almost surely unstable, shifting the interpretation of reconstruction failures from purely numerical to intrinsic. The paper also offers useful empirical evidence—correlation between the instability coefficient and reconstruction error in Stable Diffusion 3.5 and FLUX, including with a second-order Heun solver—and a universal approximation result for the mixture-of-Gaussian-neighbors class. However, the central asymptotic theorem contains a load-bearing proof gap and rests on an unrealistic distributional premise, and the supporting Theorem 3.1 has an inequality-direction problem. The empirical correlation is suggestive but does not by itself establish the amplification mechanism.","major_comments":[{"comment":"The proof that δ→0 in Theorem 4.3 is not established. The quantity δ is defined as π_real({x: ‖G^{-1}(x)‖² > 2n+3√(2n)}), but the tail computation in Appendix D.5 is performed under π_gen and then transferred to π_real via absolute continuity. A change of variables x=G(z) gives δ_n = ∫_{‖z‖²>2n+3√(2n)} γ(z)[p_real(G(z))/p_gen(G(z))] dz, and under Assumptions 4.3–4.4 the density p_gen can be exponentially small on most of the cube while p_real is only bounded above, so the density ratio need not be uniformly bounded in n. Absolute continuity alone provides no quantitative control, so δ→0 is equivalent to an unstated strong closeness condition between π_real and π_gen. Since P_M ≥ 1−ε−δ, the conclusion P_M→1 collapses without this step.","section":"Appendix D.5 / Eq. (11)"},{"comment":"Assumption 4.2, that the real image density p_real satisfies p_real ≥ C0 > 0 on the entire cube [0,1]^n, contradicts the low-dimensional manifold structure of natural images and is adopted precisely to force ε→0. The paper's claim that Theorem 4.3 is based on the characteristics of image data (Section 4.2.1) is therefore not supported; the theorem depends on a premise that real images fill the pixel cube. This is a modeling assumption rather than an internal inconsistency, but it means the asymptotic result does not apply to the image distributions for which the instability problem is posed.","section":"Assumption 4.2"},{"comment":"The derivation in Appendix D.2 produces an upper bound on the Euler global truncation error, E_n ≤ h M2/(2L)(e^L−1), and substituting L ≥ log C gives E_n ≤ h M2(C−1)/(2 log C). This is an upper bound, so it cannot justify the claimed lower bound U ≥ h M2(C−1)/(2 log C)·C in Eq. (5). The direction of the inequality in Theorem 3.1 is therefore unsupported; the theorem as stated does not demonstrate that the reconstruction error is bounded below by the displayed quantity.","section":"Theorem 3.1 / Eq. (5), Appendix D.2"}],"minor_comments":[{"comment":"The sentence 'In the next two subsections, we will first make reasonable assumptions ... in Section 4.1' should refer to Section 4.2, since the assumptions are discussed there.","section":"Section 4.2, page 7"},{"comment":"The header 'Proof of Theorem 4.1' is repeated for the proof of the universal approximation result; it should read 'Proof of Theorem 4.2'.","section":"Appendix D.4 header"},{"comment":"The correlation analyses report fit lines but no confidence intervals or significance tests; adding these would strengthen the empirical claim of a positive relationship.","section":"Figure 4(c) and Figure A1"}],"recommendation":"reject","confidential_remarks":"The manuscript is within the journal's scope and the empirical study is competently executed, but the central theoretical claim is not supported by the proof as written. The δ→0 step is a gap that cannot be repaired without either adding a strong closeness condition between π_real and π_gen (which would change the substance of the claim) or proving a uniform density-ratio bound that is unlikely to follow from Assumptions 4.1–4.4. I would not oppose a revised version that narrows the claim and corrects Theorem 3.1, but the current submission does not meet the bar for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is worth a look for its framing, but the central theorem does not hold up. The idea that sparsity in the generation distribution creates large Jacobians in the PF-ODE, and that those amplify numerical noise during reconstruction, is clear and plausible. The formal definition of intrinsic instability is a useful contribution, and the toy examples plus the correlation plots on SD3.5 and FLUX (with both Euler and Heun) are suggestive that something real is happening.\n\nThe soft spot is Theorem 4.3, and it is load-bearing. The proof of δ→0 in Appendix D.5 computes a chi-square tail under π_gen and then transfers it to π_real by absolute continuity plus a Kolmogorov extension limit. But absolute continuity gives no quantitative control. Under the stated assumptions we have δ_n = ∫_{||z||² > ...} γ(z) [p_real(G(z))/p_gen(G(z))] dz, and p_real/p_gen need not be bounded. Assumption 4.2 gives a positive lower bound on p_real on the cube, but Assumptions 4.3–4.4 make p_gen exponentially small on most of the cube, so the ratio can blow up. The conclusion δ→0 is essentially equivalent to a strong closeness between π_real and π_gen that is not derived from Assumptions 4.1–4.4. Since the lower bound P_M ≥ 1−ε−δ depends on δ→0, the asymptotic 'almost sure instability' is not established.\n\nAssumption 4.2 itself is also unrealistic: uniform positive density on all pixel configurations is not a description of natural images, and it is doing real work in the argument. That alone would make me skeptical of the asymptotic result even if the proof were fixed.\n\nA smaller point: Theorem 3.1 states that an upper bound U satisfies a lower bound; the statement is confusing, and the proof jumps from a directional Jacobian estimate to a bound on the global Lipschitz constant faster than I can follow.\n\nWhat is good remains good: the mechanism is illustrated concretely, the experiments are honest, and the paper does not oversell the empirical part as causal. But the theoretical claim that instability probability → 1 is unsupported as written.\n\nThis paper deserves a serious referee — the question is important and the formal setup is a reasonable first step — but it should come back with the δ→0 gap addressed and the assumptions rethought. I would not cite the theorem as established.","headline":"Good mechanism, shaky theorem: the instability intuition is plausible and the experiments are suggestive, but the δ→0 step in Theorem 4.3 quietly substitutes π_gen for π_real and the asymptotic conclusion does not follow.","tokens_in":23090,"tokens_out":2974,"would_cite":false,"duration_ms":30308,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Diffusion reconstruction is intrinsically unstable: as image dimension grows, the probability of instability tends to one.","keywords":["diffusion reconstruction","probability flow ODE","intrinsic instability","diffusion inversion","distribution sparsity","Jacobian singular values","asymptotic probability","error amplification"],"falsifier":"For a trained diffusion model on images of dimension n, estimate the geometric mean of the singular values of the generation map's Jacobian at the inverted noise $G^{{-1}}$(x) for real images x. If, for some fixed M > 1, the measured fraction of images with bar_E_G($G^{{-1}}$(x)) > M does not approach 1 as n increases, Theorem 4.3 is contradicted; a cheaper check is whether real image mass concentrates in high-density regions of pi_gen, which would keep the epsilon term bounded away from zero.","tokens_in":22011,"feed_emoji":"⚠️","tokens_out":5705,"duration_ms":57037,"temperature":0.7,"pith_summary":"Diffusion reconstruction inverts an image to noise through the probability-flow ODE and then regenerates it by solving the same ODE backward. This paper argues that even in the ideal, continuous-time setting this round trip fails in a strong sense: the generation mapping can amplify infinitesimal perturbations in the inverted noise. It defines instability through the Jacobian of the generation mapping and shows that instability is tied to the sparsity of the generation distribution, where most probability mass sits in scattered small regions and the surrounding space has low density. Under stated assumptions, the paper proves that the probability of encountering such instability tends to one as the image dimension grows, implying that numerical errors, however small, can be amplified into large reconstruction errors in high dimensions.","feed_headline":"Diffusion reconstruction gets unstable as image dimension grows","feed_subtitle":"A proof shows the probability of instability in generating from inverted noise tends to one as n grows.","key_machinery":"The central object is the intrinsic instability coefficient E_F(x,u) = ||J_F(x)u||/||u|| and its geometric average bar_E_F(x), the geometric mean of the singular values of the Jacobian J_F(x). The carrying identity is the density-ratio formula from Lemma D.1: (bar_E_{phi_1}(x))^n >= |prod_i lambda_i(J(1))| = p0(x0) / p1(phi_1(x0)), meaning that the expansion of infinitesimal volumes along the ODE flow equals the ratio of the initial Gaussian density to the generated density. Because the generation distribution pi_gen is sparse, real images often fall where p_gen is tiny, making this ratio large; probability preservation then forces large gradients in the vector field and hence large singular values. This is what converts distribution sparsity into instability and finally into reconstruction error amplification.","core_discovery":"The central claim is Theorem 4.3: for the ideal diffusion generation mapping G built from the probability-flow ODE, and for real images sampled from a distribution pi_real, the probability P_M = pi_real({x : bar_E_G($G^{{-1}}$(x)) > M}) satisfies P_M -> 1 as n -> infinity for every fixed M > 1, provided Assumptions 4.1 to 4.4 hold. Here bar_E_G is the geometric average of the singular values of the Jacobian of G, and $G^{{-1}}$ is the ideal diffusion inversion. In words: for high-dimensional image data, the generation mapping almost surely has an unbounded amplification factor at the inverted noise of a real image, so even infinitesimal perturbations are magnified. The proof splits the desired probability into two vanishing pieces: the chance that a real image avoids the low-density regions of the generation distribution, and the chance that its inverted noise falls far outside a Gaussian concentration ball.","pith_inferences":["If the asymptotic result transfers to latent diffusion spaces, where the relevant dimension is the latent dimension rather than pixel count, then even exact inversion samplers will still show uncorrectable reconstruction failure whenever the latent distribution is sparse.","A testable implication of the density-ratio argument is that training procedures which smooth the generation distribution, such as adding noise regularization or enforcing Lipschitz bounds on the vector field, should reduce the instability probability.","The same push-forward density-ratio argument applies to any generative flow with a sparse target distribution, so similar instability may appear in flow matching or other ODE-based generative models whenever the data lie on a low-dimensional manifold surrounded by low-density regions.","One could directly test the paper's mechanism by measuring whether reconstruction failures concentrate on inputs whose inverted noise falls in low-density regions of pi_gen, rather than on inputs with large discretization error."],"forward_implications":["Large reconstruction errors in diffusion models are not solely numerical artifacts; they are amplified by an intrinsic property of the probability-flow ODE generation mapping.","Under Assumptions 4.1 to 4.4, the probability that the geometric-average instability coefficient exceeds any fixed M > 1 converges to one as the dimension n tends to infinity.","The mechanism is the sparsity of the generation distribution: low-density regions of pi_gen must be images of large Gaussian regions, which requires large Jacobian singular values.","Numerical experiments on toy mixture-of-Gaussians settings and on Stable Diffusion 3.5 and FLUX show a positive correlation between measured instability coefficients and reconstruction error.","Improvements that only reduce ODE discretization error or make inversion numerically exact cannot remove reconstruction error if the underlying generation mapping remains unstable."],"supporting_citations":[{"why":"Defines the probability-flow ODE that the generation mapping G is built from.","marker":"[41]"},{"why":"Provides the flow-matching formulation of the PF-ODE vector field v(x,t) = E[Z - X | X_t = x] used throughout the paper.","marker":"[28]"},{"why":"Introduced diffusion inversion and reconstruction, the process whose inaccuracy the paper seeks to explain.","marker":"[40]"},{"why":"Represents the existing attribution of reconstruction errors to numerical discretization, the baseline explanation the paper argues is insufficient.","marker":"[47]"},{"why":"The production text-to-image model (Stable Diffusion 3.5) used for the correlation experiments between instability and reconstruction error.","marker":"[11]"},{"why":"Kolmogorov extension theorem used in the proof that the delta term vanishes for real-image distributions in infinite dimension.","marker":"[44]"},{"why":"Steiner formula used to bound the volume of neighborhoods around sparse regions in the proof that epsilon tends to zero.","marker":"[39]"}],"fun_headline_variants":["Diffusion ODE instability worsens as image dimension climbs","Instability in diffusion ODEs: probability approaches one in high dims","High-dimensional images trigger diffusion reconstruction instability","Diffusion ODEs: instability in generation becomes almost certain","Image dimension growth amplifies diffusion ODE instability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The asymptotic proof rests on Assumption 4.2, which requires that every point of the unit cube [0,1]^n is a possible real image with density at least a fixed positive amount, so that real images can land in the low-density regions of the generation distribution.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion ODE instability worsens as image dimension climbs","Instability in diffusion ODEs: probability approaches one in high dims","High-dimensional images trigger diffusion reconstruction instability","Diffusion ODEs: instability in generation becomes almost certain","Image dimension growth amplifies diffusion ODE instability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1455,"prompt_tokens":920,"completion_tokens":535,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":455}},"tokens_in":536,"tokens_out":535,"duration_ms":4727,"temperature":1.0,"reasoning_tokens":455,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:53:51.548378+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a trained diffusion model on images of dimension n, estimate the geometric mean of the singular values of the generation map's Jacobian at the inverted noise $G^{{-1}}$(x) for real images x. If, for some fixed M > 1, the measured fraction of images with bar_E_G($G^{{-1}}$(x)) > M does not approach 1 as n increases, Theorem 4.3 is contradicted; a cheaper check is whether real image mass concentrates in high-density regions of pi_gen, which would keep the epsilon term bounded away from zero.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the probability-flow ODE that the generation mapping G is built from."},{"cited_title":"Wallace, A","cited_arxiv_id":null,"evidence_quote":"Represents the existing attribution of reconstruction errors to numerical discretization, the baseline explanation the paper argues is insufficient."},{"cited_title":"Esser, S","cited_arxiv_id":null,"evidence_quote":"The production text-to-image model (Stable Diffusion 3.5) used for the correlation experiments between instability and reconstruction error."},{"cited_title":"Tao.An Introduction to Measure Theory","cited_arxiv_id":null,"evidence_quote":"Kolmogorov extension theorem used in the proof that the delta term vanishes for real-image distributions in infinite dimension."},{"cited_title":"Schneider.Convex Bodies: The Brunn-Minkowski Theory","cited_arxiv_id":null,"evidence_quote":"Steiner formula used to bound the volume of neighborhoods around sparse regions in the proof that epsilon tends to zero."}],"review_version":2}