{"id":"6c9481c8-5408-4729-83ee-b816bdcb44f0","arxiv_id":"2602.07875","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Gradients of any differentiable tabular constraint lie in the tangent space of the data manifold, enabling a single trained diffusion model to satisfy unseen conditions via inference-time guidance.","lead":"This paper shows that a tabular diffusion model trained once can be steered at inference time to satisfy new conditions—like filling missing values or enforcing age ranges—by applying gradient corrections along the data manifold. The practical payoff is a single reusable model that adapts to many constraints without retraining, with strong empirical results on eight tabular datasets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.2's exact-projection assumption is proven only in the ᾱ_t→1 limit; HARPOON applies guidance at every denoising step, and Figure 3 does not test tangency (angle with x̂0, not with the normal x_t−x̂0), so the central guarantee is not established at finite t.","rationale":"The reader already issued CONDITIONAL and identified Assumption 2 as the weakest assumption. I agree that Assumption 2 is problematic, but it is not the most load-bearing issue: the derivative identity in Theorem 3.2 follows from standard differentiability of orthogonal projections, so the proof could be repaired without changing the conclusion for curved manifolds. The deeper issue is the gap between the theorem's condition and the algorithm's actual usage. Theorem 3.1 establishes Q_t → π as ᾱ_t→1, yet Algorithm 1 uses the gradient update at every step, including steps where ᾱ_t is far from 1. The paper's empirical justification, Figure 3, does not measure tangency at all: it measures the angle with x̂0, which is unrelated to the tangent space. The central claim therefore rests on an unverified finite-t approximation. This reinforces the CONDITIONAL verdict: the algorithm may work well empirically, but its advertised theoretical guarantee is not established in the regime where it operates. My concrete test would settle this by measuring the normal component of the gradient on a known curved manifold across the full noise schedule. Since this is a strengthening of the reader's concern rather than a change of direction, the verdict remains UNCHANGED.","tokens_in":29569,"tokens_out":6858,"duration_ms":74854,"concrete_test":"On a synthetic dataset supported on a curved manifold with known tangent spaces (e.g., a 2D sphere or torus embedded in R^d), train the same MSE denoiser and run HARPOON. At each step t, compute x̂0=Q_t(x_t) and g=∇_{x_t}L_inf(x̂0,c), then measure the residual ‖P_{T_{x̂0}M0}⊥ g‖/‖g‖ using the known tangent space. Plot this residual against t across the full denoising schedule. If the residual is non-negligible for mid/high t, the finite-ᾱ_t extrapolation in §3.2 fails and Algorithm 1's updates are not tangential in the regime where it operates. For real tabular data, a weaker version is to replace Figure 3's angle with the angle between g and x_t−x̂0; this directly tests approximate orthogonality to the estimated normal direction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption (Assumption 2) is real but repairable: for a smooth manifold, the differential of the orthogonal projection at a regular point is exactly the orthogonal projection onto the tangent space, so Eq. (5) J_Q v = v_T holds for curved manifolds without assuming M0∩B_r = T_{x0}M0∩B_r. The more serious gap is that Theorem 3.2 assumes Q_t 'acts as an orthogonal projection' at every t, whereas Theorem 3.1 proves this only as ᾱ_t→1: its proof explicitly uses σ_t^2 = 1−ᾱ_t → 0, i.e., t→0. HARPOON (Algorithm 1, lines 6–9) applies the tangential update at every denoising step, including large t. The only empirical support for finite-t validity is Figure 3, but that figure plots the angle between ∇_{x_t}L_inf(x̂0,c) and x̂0, not between the gradient and a normal direction. For the claim 'without leaving the manifold', the gradient must be orthogonal to the normal component x_t−x̂0 (or lie in T_{x̂0}M0). Orthogonality to x̂0 is neither necessary nor sufficient for tangency. Thus the central guarantee is unverified in the actual operating regime of the algorithm, independent of the flat-geometry issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes HARPOON, a training-free conditional tabular diffusion method. The main theoretical claim is that a standard MSE-trained denoiser acts, in a suitable limit, as an orthogonal projection onto the data manifold, and that the gradient of any differentiable inference-time loss computed through this denoiser lies in the tangent space of that manifold. On this basis, the authors design an inference-time guidance algorithm that interleaves unconditional denoising with tangential gradient corrections, and evaluate it on imputation and inequality-constraint tasks across eight tabular datasets. The paper also argues that this generalises prior manifold-guidance results, which were restricted to squared-error losses and flat manifolds.","tokens_in":29921,"tokens_out":4578,"duration_ms":49364,"significance":"If the theoretical claims were established at the level stated, the paper would make a useful contribution: it would extend manifold-guided diffusion from squared-error inverse problems to arbitrary differentiable objectives, and would address a real limitation of training-time conditional tabular models. The paper is commendably explicit about its assumptions, provides proofs in the appendix, gives a reproducible experimental setup, and evaluates on a broad set of baselines and tasks. The empirical gains for categorical imputation and inequality constraints are plausible. However, the central derivation contains a false geometric assumption, an incorrect posterior in the proof of Theorem 3.1, and a finite- t gap between the theorem and the algorithm. These issues are load-bearing for the paper's main claim that HARPOON's updates stay on the data manifold, so the contribution is only partially supported as written.","major_comments":[{"comment":"Assumption 2 states M0 ∩ B_r(x0) = T_{x0}M0 ∩ B_r(x0) for some r>0. For any genuinely curved manifold this equality is false for every r>0, so the assumption contradicts the paper's advertised generality to 'curved geometries'. The proof of Theorem 3.2 (Appendix B) uses this assumption to justify Eq. (5), J_Q v = v_T. As written, the tangent-space guarantee is therefore proved only for locally flat manifolds. The assumption is repairable: for a smooth manifold, the differential of the orthogonal projection at a regular point is the projection onto the tangent space, so Eq. (5) can be obtained without the false equality. But this repair is not in the manuscript.","section":"Section 3.1, Assumption 2"},{"comment":"The proof states that under the Gaussian noising framework, p(x0|xt,t) ∼ N(x0|xt, σ_t^2 I). This is not the posterior implied by Eq. (1). From xt = sqrt(αbar_t)x0 + sqrt(1−αbar_t)ϵ, the likelihood is p(xt|x0) ∝ exp(−||xt − sqrt(αbar_t)x0||² / (2(1−αbar_t))), so the posterior is proportional to N(xt; sqrt(αbar_t)x0, (1−αbar_t)I)p(x0), not to N(xt, σ_t²I). The subsequent Gaussian localisation argument using Lemma D.9 of Stanczuk et al. is therefore not justified as written. Since Theorem 3.1 is the only formal support for the orthogonal-projection assumption in Theorem 3.2, this gap is central.","section":"Appendix B, proof of Theorem 3.1"},{"comment":"Theorem 3.1 establishes lim_{αbar_t→1} Q_t(x_t)=π(x_t), i.e. a statement only in the limit t→0. Theorem 3.2 assumes that Q_t acts as an orthogonal projection at every t, but that is not proved. Algorithm 1 (lines 6–9) applies the tangential update at every denoising step, including large t. The paper's empirical support for finite t is Figure 3, but that figure plots the angle between ∇_{x_t}L_inf(x̂0,c) and x̂0. Tangency of a vector to T_{x̂0}M0 is not equivalent to orthogonality to x̂0; the relevant normal direction is x_t − x̂0 (or the component normal to the tangent plane). The displayed angle is neither necessary nor sufficient for tangency, so the central 'without leaving the manifold' guarantee is unverified in the operating regime of the algorithm. The text already concedes that strict orthogonality holds only in the limit, which makes the gap explicit.","section":"Section 3.2 / Algorithm 1 / Figure 3"},{"comment":"After establishing J_Q v = v_T, the proof derives J_Q^T w = w_T by 'matching terms' in the equation v_T^T J_Q^T w + v_N^T J_Q^T w = v_T^T w_T. This inference does not follow from the displayed equation alone: the decomposition of w into w_T + w_N is not introduced, and the conclusion requires J_Q = P_T, which is exactly what Eq. (5) gives for all v only if J_Q is the orthogonal projection onto the tangent space. The algebra should be rewritten to show that J_Q^T = P_T first. As written, the proof is not self-contained at this step.","section":"Appendix B, proof of Theorem 3.2"}],"minor_comments":[{"comment":"The caption says 'Avg. angle between gradients and dirty estimates', which is the quantity that cannot support tangency. Consider instead reporting the angle between the gradient and the normal component x_t − x̂0, or the projection residual ||(I−P_T)∇L|| / ||∇L||, for several t.","section":"Figure 3 / Section 4.1"},{"comment":"The displayed expression for F(x_t,t) contains redundant p(x_t) factors in the numerator and denominator; this is a typo that obscures the Bayes step. Please clean up the notation.","section":"Equation (4)"},{"comment":"The gradient g is computed at x_t but applied to the denoised sample x'_{t−1}. The paper justifies this by saying consecutive shells are nearly parallel, but this is stated informally. A short quantitative statement (e.g., how the tangent spaces of M_t and M_{t−1} are related) would strengthen the presentation.","section":"Algorithm 1, line 9"},{"comment":"Remark 1 says 'our theoretical derivations provide a way to overcome this limitation for tabular diffusion', but no such derivation is given in the paper. Either provide the argument or soften the claim.","section":"Remark 1 / Section 3.1"},{"comment":"The claim that Table 4 'directly verifies Theorem 3.2' is an overstatement: the table shows that MAE/CE losses improve imputation scores, but it does not measure tangency of gradients. Please rephrase to say the results are consistent with, but not a direct verification of, the theorem.","section":"Remark 2 / Section 4.4"}],"recommendation":"major_revision","confidential_remarks":"The core idea is attractive and the experimental program is substantial, but the theory as written does not support the advertised guarantee. The false posterior in the proof of Theorem 3.1 and the flat-manifold Assumption 2 are repairable, and the finite-t gap can likely be addressed by either a stronger theoretical statement or by adding a direct empirical measure of tangency. I would not reject, but the authors need to fix the proof and add evidence specifically targeting the finite-t regime before the central claim can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a useful paper for practitioners, and the experiments are better than the theory. The genuinely new piece is the claim that with an MSE-trained denoiser, the gradient of any differentiable inference-time loss with respect to the noisy input is tangent to the data manifold, and HARPOON uses that to handle imputation and inequality constraints without retraining. The empirical work is solid: eight datasets, multiple missingness mechanisms, standard baselines, ablations on loss and schedule, and code released. I believe the method is worth engaging with.\n\nThe soft spots are in the theory section, and they matter. Theorem 3.1 proves the dirty estimate becomes the orthogonal projection only as alpha_bar_t -> 1, i.e., t -> 0. The algorithm applies the tangential update at every denoising step including large t. The paper acknowledges this and leans on Figure 3 to say the property extends in practice. But Figure 3 plots the angle between the gradient and the dirty estimate x_hat0. Tangency would require orthogonality to the normal direction x_t - x_hat0, or to a basis of the normal space at x_hat0. The figure does not establish the claim in the operating regime.\n\nAssumption 2 states the manifold equals its tangent plane in a ball. That is false for any genuinely curved manifold for every radius, so it contradicts the advertised curved-geometry generality. The stress-test note is right that this part is repairable: for a smooth manifold, the derivative of the orthogonal projection at a regular point is exactly projection onto the tangent space, so Eq. (5) can hold without the flatness assumption. But as written, the proof is internally inconsistent. The limiting theorem is also less informative than it looks: as alpha_bar_t -> 1, x_t itself approaches M0, so the limit cannot by itself justify behavior at nonzero noise.\n\nThe citation pattern is fine. The self-citation (Shankar et al. 2024) is peripheral, and the central claim does not depend on it. No circularity, no fitted constants masquerading as predictions. The method is structurally similar to prior training-free guidance (the paper admits this), and the new theoretical scope is narrower than claimed: arbitrary differentiable losses only under an unproven projection assumption at finite t. Still, the empirical gains are real, and the ablations are informative, especially the result that sparsity-inducing MAE at inference beats MSE.\n\nWho this is for: anyone building inference-time conditional tabular generation and wanting a practical recipe. It deserves a serious referee. The path forward is to replace Assumption 2 with a differentiable-projection condition, prove or bound the finite-t error, and replace Figure 3 with a test measuring the normal component explicitly. I would send it to peer review, expecting heavy revision rather than rejection. I would cite the algorithmic and empirical part in my own work; I would not cite the theorem as stated.","headline":"Solid empirical paper on inference-time tabular conditioning; the theory overclaims because the tangency guarantee is proved only in the t→0 limit and Figure 3 does not test finite-t tangency.","tokens_in":30411,"tokens_out":4910,"would_cite":true,"duration_ms":55319,"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":"A standard tabular diffusion model trained once on squared-error loss can be conditioned at inference on any differentiable constraint, because the constraint's gradient points along the data manifold rather than off it.","keywords":["tabular diffusion","manifold guidance","inference-time conditioning","imputation","inequality constraints","tangent space","orthogonal projection","training-free guidance"],"falsifier":"Train a diffusion model on points sampled from a genuinely curved manifold, such as a sphere embedded in three-dimensional space, then measure the angle between the gradient of an arbitrary differentiable loss and the manifold's tangent plane at the dirty estimate across the full denoising trajectory; if the angle deviates substantially from 90 degrees at any step, the tangent-space guarantee fails outside the flat case. A more direct check is to compute the derivative of the dirty-estimate mapping applied to a tangent vector and compare it with the vector itself—the proof requires exact equal","tokens_in":29404,"feed_emoji":"🧭","tokens_out":7288,"duration_ms":66846,"temperature":0.7,"texified_at":"2026-08-05T20:52:51.910879+00:00","pith_summary":"The paper sets out to show that conditional generation for tabular data need not be tied to training-time conditioning. Its central claim is that a diffusion model trained with the ordinary squared-error objective learns a denoising map that acts like an orthogonal projection onto the data manifold, and that the gradient of any differentiable user-supplied loss computed through that projection lies in the manifold's tangent space. On that footing the authors build HARPOON, a sampling procedure that alternates unconditional denoising steps with tangential gradient corrections, enabling imputation, inequality constraints, and mixed categorical/continuous conditions to be enforced purely at inference. If the geometric claims hold, a single pretrained model could serve arbitrary unseen constraints, a practical step beyond methods that require retraining or rejection sampling.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":3093,"prompt_tokens":746,"completion_tokens":2347,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":746,"completion_tokens_details":{"reasoning_tokens":1695}},"feed_headline":"One diffusion model steers itself to satisfy any tabular condition","feed_subtitle":"Training-time conditioning becomes unnecessary: imputation, inequalities, and categorical matches are enforced purely during sampling.","key_machinery":"The central object is the dirty-estimate mapping Q_t, defined by reversing the forward noising step: $Q_t(x_t) = \\frac{x_t - \\sqrt{1-\\bar{\\alpha}_t} \\epsilon_\\theta(x_t,t)}{\\sqrt{\\bar{\\alpha}_t}}$. The paper proves this map converges to the orthogonal projection onto the data manifold and identifies its Jacobian with the identity on tangent vectors. That Jacobian identity is the mechanism that forces gradients of arbitrary inference-time losses into the tangent space. The algorithm then uses these tangential gradients as corrections interleaved with standard denoising, with task-specific losses such as sparsity-inducing absolute error for imputation and ReLU-based soft penalties for inequalities.","core_discovery":"The paper's core discovery is a pair of theoretical statements about diffusion models. Theorem 3.1 says that, in the limit as the noise level approaches zero, the model's 'dirty estimate' of the clean sample converges to the orthogonal projection of the noisy input onto the data manifold. Theorem 3.2 then says that if the dirty estimate is an orthogonal projection, the gradient of any differentiable inference-time loss with respect to the noisy input lies in the tangent space of the manifold at the projected point. The authors use the second statement to justify an algorithm that takes gradient steps along the manifold surface while denoising, and they report that this reduces constraint vio","pith_inferences":["The theorems as stated carry a hidden caveat: the proof of tangent-space alignment relies on exact local flatness of the manifold, which no curved manifold satisfies. A natural repair is to replace the equality with a bound from the manifold's second fundamental form and quantify the resulting angle error; the paper's angle plots suggest the error may be small, which deserves direct testing.","If the orthogonal-projection behaviour holds beyond the limiting regime (which the paper verifies only empirically), the same inference-time guidance recipe should transfer to non-tabular mixed discrete/continuous modalities, such as molecular or event-stream data, where constraints are equally diverse.","The disjunctive-constraint loss exposes a possible failure mode: when the feasible region is disconnected, gradient guidance may systematically favour one branch. A testable extension is to add a diversity-promoting term and measure both violation rate and branch coverage.","Because the algorithm reuses the gradient computed at step t for the correction at step t-1, it implicitly assumes consecutive manifold shells are nearly parallel; this could be checked by measuring the angle between tangent planes at successive dirty estimates, and a large angle would motivate recomputing gradients after each denoising step."],"forward_implications":["A model trained once can be repurposed for any number of new conditions at inference, since guidance depends only on a differentiable loss evaluated on the dirty estimate.","Inference-time losses can be chosen to suit the data type—for example, sparsity-inducing MAE for one-hot encoded categories—and can outperform the training loss, removing the need to train separate discrete and continuous diffusion branches.","For inequality, categorical, conjunctive, and disjunctive constraints, the method reports violation rates below 20% on the tested benchmarks, with the lowest violations on disjunctions because the feasible region is larger.","The approach remains practical: sampling cost stays within about twice the unconditional diffusion baseline, and under five seconds on the largest tested dataset.","The tangent-space guarantee applies to any differentiable loss, so future constraint types beyond imputation and inequalities—such as domain-specific physical or regulatory bounds—can be plugged in without changing the model or its training."],"fun_headline_variants":["Manifold-guided diffusion: hit any tabular constraint at inference","No retraining needed: Harpoon steers tabular diffusion on the manifold","Condition tabular diffusion on the fly with manifold geometry","Harpoon: inference-time conditioning for tabular diffusion, no retraining","Diffusion follows the manifold to satisfy tabular conditions during sampling"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof of the tangent-space guarantee assumes that, inside a small ball around every clean data point, the data manifold coincides exactly with its tangent plane; that is true only for flat manifolds, so the theorem as proven does not cover the curved geometries the paper advertises.","fun_headline_variants_meta":{"raw":{"variants":["Manifold-guided diffusion: hit any tabular constraint at inference","No retraining needed: Harpoon steers tabular diffusion on the manifold","Condition tabular diffusion on the fly with manifold geometry","Harpoon: inference-time conditioning for tabular diffusion, no retraining","Diffusion follows the manifold to satisfy tabular conditions during sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000484,"raw_usage":{"total_tokens":2190,"prompt_tokens":674,"completion_tokens":1516,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":418,"completion_tokens_details":{"reasoning_tokens":1426}},"tokens_in":418,"tokens_out":1516,"duration_ms":12626,"temperature":1.0,"reasoning_tokens":1426,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T03:26:21.644645+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train a diffusion model on points sampled from a genuinely curved manifold, such as a sphere embedded in three-dimensional space, then measure the angle between the gradient of an arbitrary differentiable loss and the manifold's tangent plane at the dirty estimate across the full denoising trajectory; if the angle deviates substantially from 90 degrees at any step, the tangent-space guarantee fails outside the flat case. A more direct check is to compute the derivative of the dirty-estimate mapping applied to a tangent vector and compare it with the vector itself—the proof requires exact equal","supporting_citations":[],"review_version":1}