{"id":"f58f32dc-5049-48c6-ba62-919a27dc4415","arxiv_id":"2412.16748","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A diffusion inverse problem solver based on treating the reverse sampling process as an optimal control episode with iLQR, claiming recovery of posterior sampling and state-of-the-art image reconstruction.","lead":"This paper reframes diffusion-based inverse problem solving as a discrete optimal control task, using an iterative Linear Quadratic Regulator to steer a pretrained diffusion model toward reconstructions matching a measurement. It reports strong results on FFHQ 256x256 and claims the idealized posterior sampling update can be recovered as a special case, though the theoretical derivation contains gaps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The theoretical core is not established: Theorem 4.1's proof has an alpha vs 1/alpha and Riccati algebra error, so Lemma 4.2's recovery of Eq. (9) does not follow, and the implemented algorithm uses different costs and control mode than the theorem.","rationale":"The reader's weakest assumption already flagged the alpha versus 1/alpha inconsistency and the mode mixing in the Quu formulas; my independent reading of the proof confirms both and adds two concrete algebraic failures: the terminal Hessian is not propagated correctly into the first Tikhonov inverse, and Eq. (56) is arithmetically wrong. This matters because the paper's novel theoretical contribution is precisely the claim that iLQR guidance equals posterior sampling under a known control law. If Theorem 4.1 is not a proof, Lemma 4.2 is unsupported, and the abstract's statement that the idealized posterior sampling equation can be recovered as a special case is not established. The theorem-to-implementation mismatch is also load-bearing: even a corrected derivation under output-perturbation mode with negative log-likelihood terminal cost would not justify the input-perturbation, L2-fidelity, alpha = 1e-4 algorithm that produced the headline FFHQ numbers, and the headline comparisons use NFE 2500 versus 1000 for most baselines. These issues reinforce the reader's CONDITIONAL verdict rather than overturning it: the control framing is promising and potentially salvageable, but the paper currently does not support its central theoretical or empirical claims as stated. A targeted re-derivation of one backward pass is sufficient to determine whether the proof is a repairable typo or a substantive error, so no flat rejection is warranted pending that check.","tokens_in":18785,"tokens_out":10468,"duration_ms":91043,"concrete_test":"Re-derive one backward iLQR pass in the exact Theorem 4.1 setup (output perturbation, ell_0 = -log p(y|x0), ell_t = 0) for a scalar linear-Gaussian inverse problem, including the terminal Hessian ell_0,xx and Tikhonov alpha, and compare the resulting u_1 against Eq. (29). If the correct Riccati formulas give u_1 = -(V'_xx + alpha I)^-1 * grad log p(y|x0), or any expression other than alpha * grad log p(y|x0), or if substituting alpha = 1/(g(t)^2 * Delta t) does not make Eq. (18) equal Eq. (9), then the central recovery claim fails and the paper must be revised to state the actual control law.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 4's central claim is Theorem 4.1 and Lemma 4.2, but the proof is internally inconsistent. The theorem statement Eq. (29) says ut = alpha * grad_xt log p(y|x0), while the proof concludes, in Eq. (64), ut = (1/alpha) * grad_xt log p(y|x0). With Lemma 4.2's alpha = 1/(g(t)^2 * Delta t), these differ by a factor (g(t)^2 * Delta t)^2, and the printed recovery equation Eq. (65) has a sign/dimension issue: its left side is a state and its right side is a velocity, so the claimed reduction to posterior sampling Eq. (9) is not established. More fundamentally, the Riccati step is wrong: for output-perturbation mode, Quu = ell_uu + V'_xx (+ alpha I), not h_x^T V_xx h_x + alpha I as used in Eq. (58). At the first backward step, V'_xx is the terminal Hessian ell_0,xx, not zero, so the cancellation in Eqs. (52)-(53) is invalid. Eq. (56) also drops a factor of 2: h_x^T V'_x + h_x^T V'_xx (V'_xx)^-1 V'_x equals 2 h_x^T V'_x, not h_x^T V'_x. Thus Theorem 4.1 does not prove the claimed control law, and Lemma 4.2 does not recover posterior sampling. In addition, even a corrected theorem would not cover the reported method: Algorithm 1 and Table 2 use input-perturbation mode, terminal cost ||A(x0) - y||, running cost alpha * ||ut||, and alpha = 1e-4, whereas Theorem 4.1 assumes output-perturbation mode, terminal cost -log p(y|x0), zero running costs, and the Lemma's alpha. The theoretical bridge is therefore missing exactly where the experiments run.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes 'diffusion optimal control', a method for solving inverse problems by treating the discretized reverse diffusion process as an iLQR optimal control problem, with a pretrained unconditional diffusion model as the dynamics and a task-dependent terminal cost. The authors claim that Tikhonov-regularized iLQR yields controls equal to the conditional score (Theorem 4.1), that an appropriate choice of the regularization constant recovers posterior sampling (Lemma 4.2), that an analogous input-perturbation result holds (Theorem 4.3), and that the resulting solver achieves state-of-the-art image reconstruction on FFHQ 256x256 across super-resolution, inpainting, and deblurring, with additional class-conditional MNIST experiments.","tokens_in":19287,"tokens_out":11415,"duration_ms":97957,"significance":"If correct, this would be a valuable zero-shot inverse-problem solver: it avoids training conditional score models, works with any differentiable forward operator, and uses an off-the-shelf unconditional diffusion prior. The paper also ships code, includes ablations over rank, alpha, and T, and evaluates against many baselines. I do not see a circularity problem: the method is an optimization-based solver with a data-fidelity terminal cost, and the posterior-recovery statement is a claimed derived property rather than an input assumption. However, the central theoretical derivation has several load-bearing algebraic inconsistencies, and the implemented algorithm is not the one analyzed in the theorems; the empirical results are interesting but do not by themselves establish the advertised theoretical bridge.","major_comments":[{"comment":"The statement of Theorem 4.1 (Eq. 29) asserts u_t = alpha * grad_{x_t} log p(y|x0), but the proof concludes in Eq. (64) with u_t = (1/alpha) * grad_{x_t} log p(y|x0). This is not a notational slip: with the value of alpha chosen in Lemma C.1, alpha = 1/(g(t)^2 Delta t), the two expressions differ by a factor (g(t)^2 Delta t)^2. Since Lemma C.1 substitutes the theorem's control into Eq. (18), the claimed recovery of posterior sampling does not follow from the proof as written.","section":"Appendix C, Theorem 4.1"},{"comment":"The Riccati step conflates the input- and output-perturbation formulas. For output-perturbation dynamics (Eq. 18) h_u = I, so Q_uu = l_uu + V'_xx, not h_x^T V'_xx h_x + alpha I as used in Eq. (58). In addition, at the first backward sweep V'_xx is the terminal Hessian d^2 l_0/dx_0^2, generically nonzero, so the cancellation in Eqs. (52)-(53) is invalid. Eq. (56) is also arithmetically wrong: h_x^T V'_x + h_x^T V'_xx (V'_xx)^-1 V'_x equals 2 h_x^T V'_x, not h_x^T V'_x. The induction therefore does not establish that V_xx vanishes or that the control law holds.","section":"Appendix C, Eqs. (51)-(58)"},{"comment":"Equation (65) is dimensionally inconsistent: the left side is a state x_{t-1}, while the right side is a velocity-like expression without the state x_t and without the structure of Eq. (18). Even setting that aside, substituting alpha = 1/(g(t)^2 Delta t) into Eq. (18) yields a control contribution proportional to (g(t)^2 Delta t) grad log p(y|x0), not the -1/2 g(t)^2 Delta t grad log p(y|x0) term needed to turn Eq. (3) into the posterior update of Eq. (9). The sign inconsistency between Eq. (3), which has a minus before the bracket, and Eq. (9), which has a plus, compounds the problem. Lemma C.1 therefore does not demonstrate the claimed recovery of posterior sampling.","section":"Appendix C, Lemma C.1 and Eq. (65)"},{"comment":"The reported experiments do not instantiate the analyzed controller. The theorems assume output- or input-perturbation dynamics with zero running cost and a log-likelihood terminal cost, while Algorithm 1 and Table 2 use input-perturbation mode, running cost alpha ||u_t||, terminal cost ||A(x0)-y||, alpha = 1e-4, rank-1 projected second-order terms, and Adam-preconditioned updates. Theorem 4.3 is the only input-mode result, but it still assumes zero running cost and terminal cost +log p(y|x0), and its proof has the same 1/alpha and sign issues. Thus the theoretical results do not cover the method whose performance is reported, and no separate analysis of the implemented cost is given.","section":"Section 4, Algorithm 1, Table 2"},{"comment":"The claim that x0 is computed exactly on every step is overstated. The forward rollout uses the learned score network s_theta(x_t,t) inside h (Eq. (3)), so the computed x0 is exact only if the score model is exact; otherwise it inherits score-approximation error. The subsequent claim that the method is robust to score error is an empirical observation, not a consequence of exact x0 computation, and the abstract's statement that the method sidesteps dependence on score approximation needs to be qualified.","section":"Section 4, Figure 2"}],"minor_comments":[{"comment":"Equation (2) contains the typo 'dxt = dxt + ...'; the first term should be the drift, not dxt.","section":"Section 2.1, Eq. (2)"},{"comment":"The notation 'JT = 1X t=T ...' is confusing; the summation should be written explicitly as a sum over t = T, ..., 1.","section":"Section 2.3, Eqs. (10) and (34)"},{"comment":"The sign structure of Eq. (9) is inconsistent with Eq. (3): if Eq. (3) is the Euler discretization of the PF-ODE, the conditional Euler step should have the same minus-before-bracket form, but Eq. (9) has a plus. This inconsistency affects the reading of Lemma C.1.","section":"Section 2.1, Eqs. (3) and (9)"},{"comment":"The row 'Ours (NFE = 2500)' uses 2.5 times the NFE of most baselines; presenting a matched-NFE comparison in the main table, or referring more prominently to the budget analysis in the appendix, would make the performance claims easier to interpret fairly.","section":"Table 1"},{"comment":"The statement that vector-Jacobian products 'have runtime complexity O(1)' is imprecise; the intended point is that full Jacobians need not be materialized, but the actual arithmetic cost is not O(1) in dimension.","section":"Appendix D.1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: my report is deliberately stricter than the initial 'conditional' verdict because the central proof errors are not isolated typos: the wrong Q_uu formula, the alpha versus 1/alpha mismatch, and the dimensionally inconsistent Eq. (65) jointly remove the theoretical support for the paper's headline claims. The empirical method may still be valuable, but as currently written the paper does not establish the advertised posterior-recovery result, and the experiments are not covered by the theory. I would require a fully rewritten theoretical section that actually covers Algorithm 1, or a clear reframing of the theoretical claims, before considering acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the algorithmic idea is real and the FFHQ results are worth a look, but the paper's central theorem is not proved as written, and the implementation does not match the theory. Treat Section 4's posterior-sampling claims as conjecture until the derivation is redone.\n\nWhat's genuinely new: applying iLQR at inference time to a pretrained unconditional diffusion model for zero-shot inverse problems is a real departure from the DPS/DDRM line. The framework handles any differentiable forward operator, no retraining, and the ablations on rank, alpha, and T are useful. The reported numbers—even the NFE=1000 row—are competitive on super-resolution and inpainting, and the random-score experiment is a clever stress test of the method's reliance on the prior.\n\nThe soft spots are significant. The stress-test note is accurate on my reading. Theorem 4.1 states u_t = alpha * grad log p(y|x0), but the proof ends at u_t = (1/alpha) * grad log p(y|x0). Lemma 4.2's Eq. (65) as printed is dimensionally wrong: a state on the left, a velocity on the right. The Riccati step for output perturbation mode uses Quu = h_x^T V_xx h_x + alpha I, which is not the correct expression for that mode, and the cancellation in Eqs. (52)-(53) assumes V_xx^0 = 0 when it generally is not. These are not cosmetic typos; they break the claimed reduction.\n\nJust as important, the theorem does not cover the reported method. Algorithm 1 and Table 2 use input-perturbation mode, terminal cost ||A(x0)-y||, running cost alpha*||u||, and alpha = 1e-4; Theorem 4.1 is about output-perturbation mode, zero running cost, terminal -log p(y|x0), and a different alpha. The theoretical bridge is missing exactly where the experiments run.\n\nThe 'exact x0' claim is overstated: the forward rollout goes through a learned score network, so the computed x0 inherits score approximation error. The robustness experiments show the method tolerates that error, which is interesting, but 'exact' is too strong.\n\nOn compute: the headline row runs at 2500 NFE versus 1000 for baselines. The paper does include a 1000-NFE row, so it is not hiding the comparison, but the 'new baseline' phrasing should carry that caveat.\n\nWho benefits: researchers working on zero-shot diffusion inverse solvers who want an alternative to probabilistic conditioning, and anyone interested in iLQR as a practical inference-time tool. The paper deserves a serious referee—the idea is promising and the empirical work is real—but only if the referee insists on a corrected derivation and a theorem that matches the implementation.\n\nMy recommendation: send to review, not desk reject, with the expectation of major revision before acceptance.","headline":"A genuine algorithmic contribution with strong FFHQ numbers, but the proof of the posterior-sampling recovery is internally inconsistent and the theory does not match the implemented method.","tokens_in":19769,"tokens_out":4023,"would_cite":false,"duration_ms":32793,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Framing the reverse diffusion process as an iLQR optimal-control episode replaces the intractable conditional likelihood and approximate score guidance with a gradient propagated through a full rollout, and reports state-of-the-art…","keywords":["diffusion models","inverse problems","optimal control","iLQR","posterior sampling","image reconstruction","zero-shot restoration","score-based generative models"],"falsifier":"Run the method on a low-dimensional linear inverse problem with a known Gaussian prior and known posterior, compute the iLQR control $u_t$ numerically, and compare it with $\\alpha \\nabla_{x_t} \\log p(y|x_0)$ across values of $\\alpha$: Theorem 4.1 predicts an exact proportional match, so any mismatch in scaling or sign would falsify the derivation.","tokens_in":18542,"feed_emoji":"🖼️","tokens_out":7043,"duration_ms":55665,"temperature":0.7,"pith_summary":"This paper argues that inverse problems—recovering an image from a lossy measurement—can be solved by treating the discretized reverse diffusion process as an optimal control problem instead of a probabilistic sampling problem. With an iLQR controller and a terminal cost equal to the negative log-likelihood of the measurement, the control signal is exactly proportional to the gradient of the measurement likelihood with respect to the noisy state, and with a specific Tikhonov parameter it reproduces the idealized posterior sampling equation. On a 256×256 face dataset, the method reports lower FID and LPIPS than existing diffusion-based inverse solvers across super-resolution, inpainting, and deblurring tasks, and it remains stable with far fewer diffusion steps.","feed_headline":"Optimal control beats diffusion posterior samplers on image repair","feed_subtitle":"Framing reverse diffusion as an iLQR control episode avoids the intractable conditional score and wins on FFHQ 256×256.","key_machinery":"The carrying object is the discretized reverse diffusion map $x_{t-1} = h(x_t, u_t)$, together with the iterative Linear Quadratic Regulator (iLQR) algorithm. iLQR propagates a quadratic approximation of the value function backwards through the rollout; with zero running costs and a terminal cost given by the negative measurement log-likelihood, the second-order terms vanish and the feedforward gain reduces to $-\\frac{1}{\\alpha}V_x$, so the control becomes the likelihood gradient. The two ways of injecting control—input perturbation, where the control is added before the diffusion step, and output perturbation, where it is added after—produce the same structure, and the Tikhonov regularizer $\\alpha I$ on the $Q_{uu}$ inverse is the knob that matches the idealized posterior sampler. The paper also adds randomized low-rank approximations and matrix-free products to make the high-dimensional Jacobians and Hessians tractable.","core_discovery":"The central claim is that the drawbacks of diffusion-based inverse solvers—intractable conditional likelihood, dependence on score-network quality, and poor $\\mathbf{x}_0$ prediction—are not inherent to diffusion sampling but to the probabilistic framing. By defining the discrete-time reverse diffusion trajectory as the dynamics of an optimal control episode with zero running costs and terminal cost $-\\log p(y|x_0)$, the iLQR update with Tikhonov regularization produces controls $u_t = \\alpha \\nabla_{x_t} \\log p(y|x_0)$ (Theorem 4.1); with $\\alpha = 1/(g(t)^2 \\Delta t)$ and the deterministic sampler this coincides with the idealized posterior sampling equation. The same construction in input-perturbation mode yields a predictor-corrector sampler (Theorem 4.3). Empirically, the paper reports a new baseline in reconstruction performance on FFHQ 256×256 for super-resolution, random and box inpainting, Gaussian deblurring, and motion deblurring, and demonstrates that the method continues to produce feasible solutions even with a randomly initialized score network, where diffusion posterior sampling fails.","pith_inferences":["Beyond the paper: the same iLQR controller could steer toward differentiable objectives other than measurement likelihood, such as perceptual or semantic scoring functions, inside a single rollout.","Beyond the paper: applying this construction to distilled or quantized diffusion models would test whether the reported tolerance to score error survives when the prior itself is only approximately preserved.","The paper leaves open whether rank-one Hessian approximations preserve the posterior-sampling identity for strongly non-Gaussian terminal costs; a direct numerical check in a low-dimensional setting would settle that."],"forward_implications":["Because the controller only needs a differentiable forward operator, the same recipe applies zero-shot to linear and nonlinear measurements, including neural classifiers.","The posterior sampling equation used by existing probabilistic solvers becomes a special case, so improvements to the control solver directly improve posterior sampling rather than requiring a new likelihood approximation.","The method's stability at small total diffusion steps means inverse-problem solvers built this way can use fewer steps, cutting computation without the usual quality collapse.","The reported tolerance to a poorly approximated score suggests the solver can work with approximate or biased priors better than probabilistic samplers can."],"supporting_citations":[{"why":"Supplies the score-based SDE and probability-flow ODE framework, including the idealized posterior sampling equation that the paper recovers as a special case.","marker":"Song et al. [2021]"},{"why":"Defines Diffusion Posterior Sampling, the main probabilistic baseline, and provides the pretrained FFHQ weights and experimental setup used for comparisons.","marker":"Chung et al. [2023a]"},{"why":"Introduces the iLQR algorithm whose Riccati equations and gain update formulas carry the control derivation.","marker":"Li and Todorov [2004]"},{"why":"Supplies the Tikhonov-regularized variant of iLQR whose regularization parameter $\\alpha$ becomes the key constant in Theorems 4.1 and 4.3.","marker":"Tassa et al. [2014]"},{"why":"Provides the randomized low-rank approximation method used to make the high-dimensional Jacobian and Hessian computations tractable.","marker":"Halko et al. [2011]"},{"why":"Gives Denoising Diffusion Restoration Models, a representative approximation-based probabilistic solver that the paper contrasts with its control-based approach.","marker":"Kawar et al. [2022]"}],"fun_headline_variants":["Image repair via optimal control beats diffusion sampling","Diffusion inverse solvers get an iLQR upgrade","Optimal control fixes diffusion inverse problems","Reframing diffusion as control improves image reconstruction","iLQR-based diffusion beats posterior samplers on FFHQ"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The recovery of exact posterior sampling assumes the deterministic ODE sampler and that the noisy state at intermediate times contains no information about the measurement beyond the clean image, and it takes the forward rollout through the learned score network as exact.","fun_headline_variants_meta":{"raw":{"variants":["Image repair via optimal control beats diffusion sampling","Diffusion inverse solvers get an iLQR upgrade","Optimal control fixes diffusion inverse problems","Reframing diffusion as control improves image reconstruction","iLQR-based diffusion beats posterior samplers on FFHQ"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1323,"prompt_tokens":954,"completion_tokens":369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":295}},"tokens_in":570,"tokens_out":369,"duration_ms":4625,"temperature":1.0,"reasoning_tokens":295,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:14:55.781717+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the method on a low-dimensional linear inverse problem with a known Gaussian prior and known posterior, compute the iLQR control $u_t$ numerically, and compare it with $\\alpha \\nabla_{x_t} \\log p(y|x_0)$ across values of $\\alpha$: Theorem 4.1 predicts an exact proportional match, so any mismatch in scaling or sign would falsify the derivation.","supporting_citations":[{"cited_title":"[2014], resulting in the action update ut = Pkt + Kt(xt − x′ t), (80) where P is the preconditioning matrix produced by the Adam optimizer","cited_arxiv_id":null,"evidence_quote":"Supplies the Tikhonov-regularized variant of iLQR whose regularization parameter $\\alpha$ becomes the key constant in Theorems 4.1 and 4.3."}],"review_version":1}