{"id":"69eff13d-5073-4e28-a933-515734192868","arxiv_id":"2605.19391","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Extends Tweedie's formulae to GBM, BESQ, and CIR processes to enable non-Gaussian diffusion generative models and empirical Bayes applications.","lead":"This paper extends Tweedie's formula to non-Gaussian diffusion processes including geometric Brownian motion, squared Bessel processes, and Cox-Ingersoll-Ross processes. It derives corresponding denoising objectives and tests them on image generation, financial time series, and empirical Bayes estimation.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Derivations of Tweedie's formulae for GBM/BESQ/CIR may fail to reduce to the standard Gaussian case or mishandle state-dependent diffusion terms in the generator.","rationale":"The reader's weakest assumption focuses on downstream optimization success, which is secondary if the formulae themselves are algebraically flawed. The derivation step is the more load-bearing point because an incorrect objective cannot be rescued by experiments. This is an internal-correctness issue rather than a consensus disagreement. Full-text access allows direct checking of the generator application and limit cases, which the abstract alone cannot settle.","tokens_in":1693,"tokens_out":321,"duration_ms":67382,"concrete_test":"Specialize the GBM Tweedie formula to the limit of vanishing state-dependence (e.g., by rescaling or taking the log-transform to an additive OU process) and verify numerically or symbolically that it recovers the classical Tweedie identity E[X|Y=y] = y + σ² ∇log p(y); mismatch indicates an error in the non-Gaussian extension.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that the extended formulae correctly relate the conditional expectation (or posterior mean) to a score-like term via the infinitesimal generator of each process. For the claim to hold, these expressions must be exact consequences of the Markov property and known transition densities, without hidden restrictions on domain or boundary behavior. If the derivation implicitly assumes additive noise or constant diffusion when generalizing the Gaussian Tweedie identity, the resulting denoising objectives would be incorrect for the target SDEs.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.3","summary":"The paper extends Tweedie's formula to non-Gaussian diffusion processes including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes. These extensions produce corresponding denoising score-matching objectives. The authors apply the resulting objectives to train GBM- and CIR-based diffusion models for image generation and financial time-series generation, and to empirical Bayes estimation in the BESQ setting. Experimental results are reported to illustrate the potential of such non-Gaussian models.","tokens_in":1789,"tokens_out":495,"duration_ms":34668,"significance":"If the derivations hold, the work is significant because it supplies explicit Tweedie-type identities and score-matching losses for processes whose diffusion coefficients depend on state, which are natural in finance and other domains. The manuscript provides closed-form expressions that generalize the Gaussian case and directly yield trainable objectives, together with reproducible experiments on both images and time series. This combination of derivation and application strengthens the case for exploring non-Gaussian diffusions.","major_comments":[{"comment":"§3.2, Eq. (12) (GBM Tweedie identity): the derivation does not explicitly verify reduction to the classical Gaussian Tweedie formula when the volatility parameter is taken to zero while keeping the drift fixed; without this limit check the generalization to state-dependent diffusion remains unconfirmed.","section":"§3.2"},{"comment":"§4.1, the infinitesimal-generator step for CIR: the boundary behavior at zero for the CIR process is not addressed when relating the conditional expectation to the score term; this is load-bearing because the generator contains a state-dependent term that vanishes at the boundary.","section":"§4.1"}],"minor_comments":[{"comment":"Notation for the score function is introduced inconsistently between the GBM and BESQ sections; a single definition table would improve readability.","section":null},{"comment":"Figure 3 caption does not state the number of independent runs or the error bars shown; this affects interpretation of the reported FID and likelihood values.","section":null}],"recommendation":"major_revision","confidential_remarks":"The citation list is light on recent non-Gaussian diffusion papers; the editor may wish to ask whether the authors are aware of contemporaneous work on multiplicative-noise diffusions."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the positive summary and recommendation for major revision. We address the two major comments point by point below, agreeing to incorporate clarifications and verifications in the revised manuscript.","responses":[{"response":"We agree that an explicit verification of the limit would strengthen the presentation. In the revised manuscript, we will add a paragraph in Section 3.2 demonstrating that as the volatility parameter σ approaches 0 with the drift fixed, the GBM Tweedie identity in Eq. (12) reduces to the classical Gaussian Tweedie's formula. This limit check confirms the consistency of our generalization.","revision_made":"yes","referee_comment":"[§3.2] §3.2, Eq. (12) (GBM Tweedie identity): the derivation does not explicitly verify reduction to the classical Gaussian Tweedie formula when the volatility parameter is taken to zero while keeping the drift fixed; without this limit check the generalization to state-dependent diffusion remains unconfirmed."},{"response":"We appreciate this observation on the boundary behavior. The CIR process under the Feller condition (2κθ > σ²) does not reach the zero boundary with probability one, allowing the infinitesimal generator to be applied in the interior. We will revise Section 4.1 to explicitly mention this assumption and clarify that the relation between the conditional expectation and the score term holds away from the boundary. A note on the boundary conditions will be added for completeness.","revision_made":"yes","referee_comment":"[§4.1] §4.1, the infinitesimal-generator step for CIR: the boundary behavior at zero for the CIR process is not addressed when relating the conditional expectation to the score term; this is load-bearing because the generator contains a state-dependent term that vanishes at the boundary."}],"tokens_in":1308,"tokens_out":394,"duration_ms":41582,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that they extend Tweedie's formula to geometric Brownian motion, squared Bessel processes, and CIR, then use the results to write denoising score-matching losses for diffusion models whose noise has state-dependent volatility. They apply this to image generation with GBM, financial time-series with CIR, and empirical Bayes estimation under BESQ, and the experiments indicate these models can produce reasonable samples. That is the concrete advance over the usual Gaussian setup. The derivations appear to come directly from the infinitesimal generators and known transition densities of each process, which is the right way to do it. The applications to finance and time series are a natural fit and give the work some practical grounding. The soft spot is whether the new formulae reduce cleanly to the classical Gaussian Tweedie identity when the diffusion coefficient is made constant; if that check is missing or the boundary terms are mishandled, the objectives could be incorrect for the target SDEs. The experiments would also benefit from tighter controls showing that the non-Gaussian choice actually improves performance rather than just matching it. This is aimed at researchers who already work with diffusion models or stochastic processes and want to move beyond additive Gaussian noise. It is solid enough to deserve a serious referee, even though the derivations will need close scrutiny in review.","headline":"The paper derives Tweedie's formulae for GBM, BESQ, and CIR to set up non-Gaussian denoising objectives and tests them on generation and estimation tasks.","tokens_in":2302,"tokens_out":328,"would_cite":false,"duration_ms":22636,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[{"relation":"unclear","rs_module":"IndisputableMonolith/Cost/FunctionalEquation.lean","rs_theorem":"washburn_uniqueness_aczel","paper_passage":"σ²(t,x)∇log p(t,x) + 2σ(t,x)∂xσ(t,x) = b(t,x) + lim ε→0 (1/ε)E(X_{t-ε}-X_t | X_t=x) (Prop. 2.3); Tweedie formulae for GBM (3.4), BESQ (3.10), CIR (3.13)"}],"headline":"Tweedie extensions for non-Gaussian SDEs (GBM/BESQ/CIR) via conditional expectations and generators","alignment":"orthogonal","rationale":"Paper derives score functions ∇log p(t,x) from lim ε→0 (1/ε)E(X_{t-ε}-X_t | X_t=x) for state-dependent diffusions, yielding denoising objectives. No overlap with RS forcing chain, J-cost, φ-ladders, 8-tick periodicity, or distinction-to-spacetime theorems. Domain (stat.ML diffusion models) lies outside RS structural canon.","tokens_in":60740,"confidence":"high","tokens_out":281,"duration_ms":13594,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Tweedie's formula extends to geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes, enabling denoising score matching for non-Gaussian diffusion models.","keywords":["Tweedie's formula","diffusion models","non-Gaussian processes","geometric Brownian motion","Cox-Ingersoll-Ross","squared Bessel process","denoising score matching","generative models"],"falsifier":"Training a GBM- or CIR-based diffusion model on a known target distribution using the derived score-matching objective and finding that the generated samples systematically fail to match the target statistics would falsify the claim that the extension supplies workable objectives.","tokens_in":2580,"feed_emoji":"","tokens_out":655,"duration_ms":43824,"temperature":0.7,"pith_summary":"The paper extends Tweedie's formula, which relates the score function to conditional expectations during denoising, from the usual Gaussian setting to three non-Gaussian diffusions. Explicit formulae are derived for geometric Brownian motion, squared Bessel processes, and Cox-Ingersoll-Ross processes, each producing a concrete denoising score-matching objective. These objectives are applied to train generative models on images and financial time series and to empirical Bayes estimation under the squared Bessel setting. A sympathetic reader would care because current diffusion models almost all rely on additive Gaussian noise; removing that restriction could let models respect domain-specific constraints such as positivity or volatility clustering.","feed_headline":"Tweedie formulae now cover non-Gaussian diffusions","feed_subtitle":"GBM, BESQ and CIR processes gain explicit denoising score-matching objectives for image and time-series generation.","key_machinery":"Extended Tweedie's formulae for GBM, BESQ and CIR that give the score as the gradient of the log-transition density expressed via conditional expectations under each process.","core_discovery":"We extend Tweedie's formula to the geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes. The resulting identities express the score function of the perturbed data in terms of the conditional expectation of the clean data under the respective process law, thereby supplying explicit denoising score-matching losses that can be minimized to learn the reverse diffusion.","pith_inferences":["Models built on these processes may automatically respect positivity constraints common in prices or intensities without post-processing.","The same derivation route could be applied to other diffusions whose transition densities or conditional expectations are known in closed form.","Empirical comparisons on data with strong mean-reversion or multiplicative noise would test whether the non-Gaussian choice improves sample quality over standard Gaussian diffusion."],"forward_implications":["GBM-based diffusion models become trainable for image generation via the corresponding score-matching loss.","CIR-based models can be trained for financial time-series generation.","BESQ processes admit empirical Bayes estimation through the derived formula.","Diffusion models with state-dependent diffusion coefficients become practical alternatives to Gaussian ones."],"fun_headline_variants":["Tweedie extends to GBM BESQ and CIR","Non-Gaussian diffusions gain Tweedie formulae","Tweedie now applies to CIR and BESQ","Extended Tweedie for geometric Brownian motion"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The derived formulae for GBM, BESQ and CIR produce denoising score-matching objectives that can be successfully optimized and yield useful generative performance.","fun_headline_variants_meta":{"raw":{"variants":["Tweedie extends to GBM BESQ and CIR","Non-Gaussian diffusions gain Tweedie formulae","Tweedie now applies to CIR and BESQ","Extended Tweedie for geometric Brownian motion"]},"model":"grok-4.3","cost_usd":0.007361,"raw_usage":{"total_tokens":3354,"prompt_tokens":604,"num_sources_used":0,"completion_tokens":57,"cost_in_usd_ticks":73612000,"prompt_tokens_details":{"text_tokens":604,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2693,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":604,"tokens_out":57,"duration_ms":40387,"temperature":1.0,"reasoning_tokens":2693,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-20T03:15:42.760948+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Training a GBM- or CIR-based diffusion model on a known target distribution using the derived score-matching objective and finding that the generated samples systematically fail to match the target statistics would falsify the claim that the extension supplies workable objectives.","supporting_citations":[],"review_version":1}