{"id":"48b4fd49-3fe4-4d03-9fcb-787b291cb887","arxiv_id":"2502.04670","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A training-free diffusion sampling method exploits an observed linear relation between initial noise perturbations and output changes to control the sample mean and diversity around a target image.","lead":"This paper shows that slightly changing the random noise fed into a diffusion image sampler changes the output roughly linearly, and uses that to sample images with a user-chosen average distance from a target picture. The authors build a training-free sampler that tunes one knob to hit a target diversity while keeping the sample mean near the target.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.1's centering step treats the Δ-dependent Jacobian as a constant and ignores the radial (a−1)xT term; output-mean control is therefore unproven and not empirically isolated.","rationale":"The reader identified the same load-bearing weakness: Section 4.1 treats A = γ0(axT + bΔ) as a constant when taking expectations, so the step E[x̂′0] = x0 + bAE[Δ] is mathematically unjustified. I agree that this is the weakest point in the argument, because it is the bridge from the well-founded local linearity (Propositions 1–3) to the paper's headline capability of sampling around a specified target mean. I add one closely related detail: even with A fixed at γ0(xT), the spherical interpolation introduces a deterministic radial input shift (a−1)xT whose output effect (a−1)γ0(xT)xT is not addressed, so the centering claim would need an additional cancellation even if the Δ-dependence of A were handled. Proposition 1 itself is a standard differentiability argument and appears correct; the empirical R² values and PSNR comparisons are meaningful evidence in the method's favor, though they lack error bars and code. The appropriate outcome is therefore not a rejection but a conditional acceptance: the authors should either prove the required cancellation under stated assumptions or provide a direct empirical test of output-mean centering with error bars. Since this matches the reader's conditional verdict, no verdict change is needed.","tokens_in":17217,"tokens_out":10285,"duration_ms":108801,"concrete_test":"Re-run Algorithm 1 on FFHQ with the paper's protocol (50 target images, 120 samples per target, C0 values including 0.1, 0.4, and the controller-selected values matching Table 2). For each target, compute the empirical mean residual μi = E[x′0] − x0 over the 120 samples and report ∥μi∥, its 95% confidence interval, and the pooled average across targets, against the target rMSE 0.12. Separately estimate the radial term by computing ∥x0(cos(C0)·xT) − x0(xT)∥ for the same targets. If the mean residuals are within, say, 10% of the target rMSE and the radial term is negligible at the used C0, the centering claim holds empirically despite the flawed derivation; if not, the central claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that CCS produces samples whose mean is x0 rests on the derivation in Section 4.1. There, the paper writes x̂′0 := x0 + bAΔ with A = γ0(axT + bΔ), then computes E[x̂′0] = x0 + bA E[Δ] and concludes that E[Δ] = 0 suffices for E[x′0] ≈ x0. This is invalid: A is a function of the random Δ, so E[AΔ] ≠ A E[Δ] in general, and the missing term E[γ0(axT + bΔ)Δ] need not vanish. A second, independent gap is that the actual input perturbation is x′T − xT = (a−1)xT + bΔ; even if the Jacobian were held fixed at γ0(xT), the first-order mean shift would include (a−1)γ0(xT)xT, which is not zero for the spherical interpolation weights (a ≈ cos C0, b ≈ sin C0). Proposition 1 is a pointwise Taylor expansion in λ and does not justify moving the Jacobian outside the expectation or dropping the a-scaling. Proposition 6 controls only the input distance ∥x′T − xT∥, not the output mean. Thus the output-mean guarantee, which is the main contribution over prior per-sample control, is not established by the provided theory. The empirical PSNR values in Table 2 could in principle support the conclusion, but no per-target mean residuals, confidence intervals, or ablations over C0 are reported, so the required cancellation is untested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies how perturbing the initial noise vector of DDIM/ODE sampling affects the generated image. It claims, both theoretically (Propositions 1-3) and empirically (Table 1), that the output changes approximately linearly with the perturbation scale. Based on this, it proposes CCS (and P-CCS for latent diffusion), which uses spherical interpolation between the inverted noise of a target mean and a random Gaussian vector, with a binary-search controller (Algorithm 2) to match a desired output diversity (rMSE). Experiments on FFHQ, CIFAR-10, CelebA-HQ, and fMoW report that CCS achieves higher PSNR between sample mean and target mean than adapted baselines while preserving quality and diversity. The paper also demonstrates an image-editing application.","tokens_in":17583,"tokens_out":3465,"duration_ms":32569,"significance":"If the linearity and centering claims are correct, CCS is a simple, training-free tool for distribution-level control of diffusion sampling, relevant to privacy, constrained generation, and editing. The reported linearity phenomenon (R^2 between 0.947 and 0.995 across four datasets) is a useful empirical observation, and the theoretical first-order expansion (Proposition 1) is a reasonable pointwise statement. However, the paper's main contribution over prior per-sample control is the claim that E[x'_0] ≈ x0, and that claim rests on a derivation gap (Section 4.1) and aggregate statistics without per-target residuals or confidence intervals. The method itself is novel and the empirical results are promising, but the central guarantee needs a corrected derivation or direct empirical validation before publication.","major_comments":[{"comment":"The derivation of E[x'_0] ≈ x0 is invalid as written. The paper defines A = γ0(a xT + bΔ) and then writes E[x̂'_0] = x0 + b A E[Δ], treating A as a constant when taking the expectation. But A is a function of the random perturbation Δ, so the second term should be b E[γ0(a xT + bΔ)Δ]. The paper neither proves that this expectation vanishes nor reports an experiment that isolates this term. This step is load-bearing for the 'close to target mean' design goal, so the central claim is not established by the provided theory.","section":"Section 4.1"},{"comment":"Even if A were treated as constant, the perturbation is x'_T − x_T = (a−1)x_T + bΔ, with a = sin(θ−C0)/sin(θ) and b = sin(C0)/sin(θ). For C0 > 0, a < 1, so the first-order output shift includes (a−1) γ0(x_T) x_T, which is not zero. Proposition 1 is a pointwise expansion in a scalar λ and does not justify dropping this radial term or moving the Jacobian outside the expectation. Consequently, the conclusion E[x'_0] ≈ x0 in Section 4.1 is not a consequence of the stated propositions.","section":"Section 4.1 and Algorithm 1"},{"comment":"Proposition 6 controls only the input distance ||x'_T − x_T||, not the output mean. The bridge from input-distance control to output-rMSE control relies on the empirically observed linearity, but the reported PSNR values in Tables 2-4 are averaged over all target means without per-target residuals, confidence intervals, or ablations over C0. Since the theoretical centering argument is incomplete, the empirical evidence should directly test the cancellation E[γ0(a xT + bΔ)Δ] ≈ 0 and the contribution of the (a−1) term; currently it does not.","section":"Section 5.2 and Proposition 6"}],"minor_comments":[{"comment":"The binary-search update rules in lines 8 and 10 appear to be mistyped: they read '(Chigh, C0) ← (C0, C0+Clow/2)' and '(Clow, C0) ← (C0, C0+Chigh/2)', which are ambiguous and would not set C0 to the midpoint of the new interval. Please rewrite the updates clearly.","section":"Algorithm 2"},{"comment":"The probability statement in Proposition 4 has a misplaced bracket: it should read P(||X||_2^2 ∈ (1±δ)d) ≥ 1 − 2 exp(...), not P(||X||_2^2 ∈ (1±δ)d]).","section":"Proposition 4"},{"comment":"The R^2 values in Table 1 are computed after per-target normalization y' = (y−b)/a using a fitted slope and intercept. This normalization removes the per-target slope information and can inflate the apparent linearity; please also report R^2 on the unnormalized residuals or the distribution of per-target slopes.","section":"Section 5.1"},{"comment":"In Step 2 of Algorithm 3, the sampling notation 'ϵ ∼ N(0, 1 − αt)' should specify the variance as (1 − α_{t0})I with matching dimensionality; as written it is ambiguous.","section":"Algorithm 3"},{"comment":"No code or reproducible implementation details are provided, which makes it difficult to verify the exact inversion procedures, especially the partial inversion for latent diffusion (P-CCS) where t0 = 45 is a tuned hyperparameter on one validation image.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper's main idea—perturbing initial noise on the sphere to control the output distribution—is appealing and the linearity observation is worth publishing. However, the theoretical justification for output-mean control (Section 4.1) contains a genuine gap that is load-bearing for the central claim. The empirical results are promising but lack per-target statistics, error bars, and code, so the gap is not yet closed by evidence. I recommend major revision rather than rejection because the issue is fixable: the authors could either provide a rigorous treatment of E[γ0(a xT + bΔ)Δ] (e.g., under a low-rank or symmetric perturbation assumption) or add direct empirical validation of the centering claim with per-target residuals and C0 ablations. Please also ask them to be more careful about novelty claims; prior work on initial-noise editing (Chen et al., Zheng et al., Wang et al.) is cited, but the 'first to investigate' phrasing overstates the novelty."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll get straight to it. This paper documents a real and useful finding: for DDIM/ODE sampling, perturbing the initial noise at small-to-moderate scales produces output changes that are approximately linear in the perturbation scale. The R-squared values (0.98+ for pixel diffusion, 0.94+ for latent diffusion) are convincing, and the visualization in Fig. 2 makes the phenomenon tangible. The proposed CCS method is admirably simple - DDIM inversion, spherical interpolation, and a binary-search controller for C0 - and the empirical results against adapted baselines are respectable. The latent diffusion extension (P-CCS) with partial inversion is pragmatic, and the image editing application is a nice bonus.\n\nBut the theoretical justification for the main claim, that the sample mean lands at the target, has a real hole. In Section 4.1, the paper defines A = gamma0(axT + bΔ) and then writes E[x0'] = x0 + b A E[Δ]. That is only valid if A is independent of Δ, which it is not. You cannot pull the random Jacobian out of the expectation. The missing term E[gamma0(axT + bΔ) Δ] needs to be shown negligible, either by analysis or direct measurement. A second, related gap: the actual perturbation to the input is xT' - xT = (a-1)xT + bΔ. Even with a fixed Jacobian gamma0(xT), the first-order mean shift includes (a-1)gamma0(xT)xT, which is generally nonzero for moderate C0. Proposition 6 only controls input distance, not output mean. So the centering guarantee, which is the paper's headline contribution over per-sample control, is asserted rather than demonstrated.\n\nThat said, the flaws are addressable. The empirical PSNR values in Table 2 suggest the mean fidelity is often good, but without confidence intervals or per-target residuals, we can't tell if the effect is small by luck or by mechanism. The theory itself is thin: Propositions 1-3 are standard differentiability and Gronwall arguments, and Proposition 5 is a trivial variance identity. The novelty lies in the combination and the empirical validation, not in the mathematics.\n\nThis is a solid paper that deserves serious peer review. A careful referee should ask for a corrected derivation or an explicit experimental check of the missing expectation term and the radial contribution. If the authors can close that gap, the paper would be a useful contribution to controllable generation. I'd send it to review rather than desk-reject, and I'd likely cite the linearity observation in my own work, though not the unproven mean-control theorem.","headline":"A genuinely useful empirical linearity result, but the paper's central mean-control claim rests on an unjustified expectation step that treats a random Jacobian as constant.","tokens_in":18054,"tokens_out":2699,"would_cite":true,"duration_ms":28226,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"DDIM diffusion sampling responds almost linearly to initial-noise perturbations, and the paper builds a sampling controller on that fact.","keywords":["diffusion models","DDIM sampling","initial noise perturbation","linear response","controllable generation","constrained sampling","spherical interpolation","sample diversity"],"falsifier":"Run the CCS algorithm around one fixed target image with many random draws at several $C_0$ values and compute the empirical sample mean of the generated images: if its distance from the target grows with $C_0$ beyond the controller tolerance (equivalently, if $E[\\gamma_0(a x_T + b\\Delta)\\Delta]$ is not small), the centering claim fails.","tokens_in":17034,"feed_emoji":"🎯","tokens_out":8837,"duration_ms":96606,"temperature":0.7,"pith_summary":"Diffusion sampling, in the deterministic DDIM form, has a simple response to small changes in the random noise it starts from: the final image moves in a nearly linear way. The paper proves an expansion $x_0(x_T + \\lambda \\Delta x, T) = x_0(x_T) + \\lambda \\gamma_0(x_T) \\Delta x + o(\\lambda)$ using only smoothness of the log-density, with an analogous statement for the probability-flow ODE, and measures near-perfect linearity ($R^2 \\ge 0.94$) on four image benchmarks. On top of this, the proposed CCS algorithm interpolates the original noise with a fresh Gaussian vector on the sphere, and binary-searches the interpolation angle so the average distance from a target image lands on a user-specified diversity level. If the linearity result holds, this gives a training-free way to enforce distribution-level constraints on diffusion samples, such as a target mean and a chosen spread, without per-sample projections that degrade quality.","feed_headline":"Diffusion samples respond linearly to initial-noise edits","feed_subtitle":"The paper's CCS controller exploits this rule to center images on a target while tuning diversity.","key_machinery":"The object carrying the argument is the Jacobian $\\gamma_0(x_T)$ of the map from initial noise to final sample. In the discrete setting it is built by composing one-step DDIM maps $L_t(x) = \\eta_t x + \\lambda_t \\nabla \\log p_t(x)$ and differentiating: $\\gamma_T(x) = \\eta_T + \\lambda_T H_T(x)$, with $\\gamma_{t-1}$ obtained by the chain rule through the recursion; smoothness of the log-density, namely existence of the Hessian, is the sufficient condition. The same Jacobian emerges in the continuous ODE from differentiability of the flow with respect to its initialization, proved with Picard-Lindelof and Gronwall arguments. The CCS algorithm then uses the spherical interpolation formula $x'_T = \\frac{\\sin(C_0)}{\\sin(\\theta)} \\epsilon + \\frac{\\sin(\\theta - C_0)}{\\sin(\\theta)} x_T$ with $\\theta = \\angle(\\epsilon, x_T)$, which preserves the Gaussian-sphere norm, and a binary-search controller that tunes $C_0$ to hit a target MSE; the linear response is what makes the binary search converge quickly.","core_discovery":"The central discovery claimed is that the input-output map of DDIM/ODE diffusion sampling is locally linear in the initial noise: Proposition 1 gives $x_0(x_T + \\lambda \\Delta x, T) = x_0(x_T) + \\lambda \\gamma_0(x_T) \\Delta x + o(\\lambda)$, and Proposition 2 gives the continuous-time analogue. The authors then claim this linearity can be exploited: by writing $x'_T = \\frac{\\sin(C_0)}{\\sin(\\theta)} \\epsilon + \\frac{\\sin(\\theta - C_0)}{\\sin(\\theta)} x_T$, the generated sample keeps high quality because the perturbed noise stays on the Gaussian hypersphere, while its mean stays near the target $x_0$; the binary-search controller picks $C_0$ so that $E[\\|x'_0 - x_0\\|_2]$ equals a target MSE. Empirically, the normalized average residual norm tracks $\\sin(C_0)$ with $R^2$ values of 0.995 on FFHQ, 0.988 on CIFAR-10, 0.959 on CelebA-HQ, and 0.947 on fMoW, and CCS achieves the highest PSNR among the compared controller-based methods on all three image benchmarks.","pith_inferences":["Beyond the paper: if the noise-to-image map is linear in this regime, then distribution-level constraints beyond a target mean, such as a target covariance or marginal moments, could be imposed by choosing the distribution of the perturbation $\\Delta$ and pushing it through the estimated Jacobian, without per-sample projections.","Beyond the paper: the same Jacobian $\\gamma_0$ could be estimated once per target image and reused to solve multi-constraint control problems with a single linear solve, replacing the binary-search controller.","Beyond the paper: the lower $R^2$ on the latent diffusion benchmarks suggests the nonlinear autoencoder and inexact classifier-free-guidance inversion are the main sources of nonlinearity; testing the same spherical perturbation in a better-inverted latent space would isolate whether the linearity is a property of the ODE sampler or of the full pipeline."],"forward_implications":["A user can specify a target image and a desired diversity level (rMSE), and the controller finds a perturbation angle $C_0$ such that the generated batch has that average distance while its sample mean stays close to the target.","The spherical-interpolation mechanism avoids the quality collapse that plain additive noise causes: Proposition 5 shows additive perturbations push the input off the Gaussian hypersphere, while Proposition 4 guarantees a standard Gaussian noise vector stays on it.","Because the linearity also holds, more weakly, for latent diffusion models through partial inversion, the method applies to text-guided generators and to controlled image editing between a source prompt and a target prompt.","The reported $R^2$ values, 0.995 on FFHQ, 0.988 on CIFAR-10, 0.959 on CelebA-HQ, and 0.947 on fMoW, quantify how predictable the distance control is on these benchmarks."],"supporting_citations":[{"why":"Defines the DDIM sampler whose recursion is the object of Proposition 1.","marker":"Song et al. (2021)"},{"why":"Supplies the probability-flow ODE formulation and the Gaussian concentration fact used in the design.","marker":"Song et al. (2020)"},{"why":"Earlier local-linearity result for the posterior mean; the paper contrasts its own noise-to-sample linearity against it.","marker":"Chen et al. (2024b)"},{"why":"Provides the ODE stability and Gronwall tools used in the proofs of Propositions 2 and 3.","marker":"Hartman (2002)"},{"why":"Supplies the spherical interpolation formula that defines the CCS perturbation mechanism.","marker":"Shoemake (1985)"},{"why":"The Stable Diffusion backbone used in the latent and conditional generation experiments, whose autoencoder limits linearity.","marker":"Rombach et al. (2022)"}],"fun_headline_variants":["Diffusion ODE linearity lets CCS steer sample statistics","CCS exploits linear noise-output map for constrained diffusion sampling","Binary-search controller hits target MSE via noise perturbation","Linear noise-output relation enables precise diffusion sampling control","CCS: precise diffusion sampling via initial noise perturbation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"When the paper averages over random perturbations, it treats the sampler's local linear response as a fixed multiplier independent of the perturbation; if that multiplier correlates with the perturbation, the sample mean will drift off the target, and the paper neither proves nor tests that correlation is zero.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion ODE linearity lets CCS steer sample statistics","CCS exploits linear noise-output map for constrained diffusion sampling","Binary-search controller hits target MSE via noise perturbation","Linear noise-output relation enables precise diffusion sampling control","CCS: precise diffusion sampling via initial noise perturbation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000374,"raw_usage":{"total_tokens":2017,"prompt_tokens":986,"completion_tokens":1031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":955}},"tokens_in":602,"tokens_out":1031,"duration_ms":7809,"temperature":1.0,"reasoning_tokens":955,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T21:55:58.360633+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the CCS algorithm around one fixed target image with many random draws at several $C_0$ values and compute the empirical sample mean of the generated images: if its distance from the target grows with $C_0$ beyond the controller tolerance (equivalently, if $E[\\gamma_0(a x_T + b\\Delta)\\Delta]$ is not small), the centering claim fails.","supporting_citations":[{"cited_title":"Ordinary differential equations","cited_arxiv_id":null,"evidence_quote":"Provides the ODE stability and Gronwall tools used in the proofs of Propositions 2 and 3."}],"review_version":1}