{"id":"ba0e3784-6227-4bed-9d63-38357f021f39","arxiv_id":"2608.08594","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Soft Denoising Diffusion Bridge Models regularize diffusion bridges by prescribing a non-degenerate Gaussian terminal marginal, yielding closed-form dynamics that avoid terminal singularities and improve image restoration.","lead":"SDDBMs replace the exact endpoint of diffusion bridges with a flexible Gaussian terminal constraint, avoiding the numerical blow-up that hard endpoints cause. The paper derives the closed-form bridge and shows it improves deraining, super-resolution, and inpainting over existing bridge models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The x0-free reformulation matches only time-marginals, not transition laws; the reverse-time score and training posterior are then assumed rather than derived, so Eq. (18) is not shown to be the reverse of the soft bridge.","rationale":"In good faith, the core fixed-x0 construction is coherent: prescribing a Gaussian terminal marginal, solving for the Gaussian weight q, obtaining the closed-form h-function, and deriving Gaussian forward marginals are all supported by explicit algebra. The unification claims in Proposition 4.1 also check out as special-case limits. I do not see an internal flaw in Theorems 3.2–3.4 themselves. The load-bearing problem is the bridge from that construction to the x0-free sampler used in experiments. The reader identified the terminal initialization x_T ≈ (b_T + c_T)x⋆ and the per-task ζ as the weakest link; I agree those are symptoms, but I locate the root earlier: the x0-free SDE is only shown to match one-time marginals, not the transition kernel or conditional bridge law. The training posterior in A.6 is constructed rather than derived from the actual h-transformed dynamics, and the score in Eq. (18) is consequently not tied to the prescribed terminal marginal. This is an internal gap between the stated path-measure theory and the sampling pipeline, not a disagreement with external consensus. A repair is plausible—for example, by deriving the correct conditional law of Eq. (38) and using actual terminal samples—but until that is done, the empirical superiority claim is conditional on unstated approximations. Since the reader already rendered a CONDITIONAL verdict, this sharper concern does not move the verdict; it reinforces it.","tokens_in":33898,"tokens_out":16357,"duration_ms":195770,"concrete_test":"Derive the true transition density of the soft h-bridge in Eq. (15) directly from its linear SDE coefficients, and compute the conditional law of x_t given (x_s, x0, x_T). Compare its conditional mean and variance with the posterior μ_{t-1}, σ^2_{t-1} used in §3.5/A.6, and compare the implied score with the x_T=x⋆ score used in Eq. (18). If they differ, re-run the Rain100H deraining experiment using (i) the actual h-bridge conditional score and (ii) terminal samples drawn from N(x*, σ^2 I) instead of x_T ≈ (b_T + c_T)x⋆. If the Table 2 margins shrink or reverse, the reported state-of-the-art results depend on the unverified approximation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Appendix E.3 explicitly states that Eq. (38) 'should be interpreted as a marginally consistent reformulation ... rather than as a pathwise identity.' That concession marks the load-bearing gap. For a Markov Gaussian process, matching the one-time marginals in Proposition 3.4 does not determine the transition density p_h(x_t | x_s, x_T), and the reverse SDE in Eq. (18) requires the conditional score of the actual forward process. Section A.6 does not compute this score from the SDE in Eq. (15); instead it 'construct[s] a Gaussian one-step forward transition that is consistent with the marginals' (Eq. 74) and derives the training posterior from that assumed transition. Different linear SDEs can share the same marginal means and variances yet have different conditional laws, so the learned score may be the score of a different process. Appendix I then compounds the gap by replacing x_T with x⋆ in the network and initializing the sampler at x_T ≈ (b_T + c_T)x⋆, while ζ is a per-task fitted rescaling of the score drift. These three steps—marginal-only reformulation, constructed posterior, and terminal substitution—are what connect Theorems 3.2–3.4 to Table 2, and none of them is derived from the prescribed terminal marginal in Eq. (10).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes Soft Denoising Diffusion Bridge Models (SDDBMs), a class of diffusion bridges in which the hard terminal condition x_T = x* is replaced by a prescribed non-degenerate Gaussian terminal marginal N(x*, σ²I) under the h-transformed path measure. The authors derive a Gaussian terminal weight q, a closed-form soft h-function (Theorem 3.3), closed-form forward marginals (Proposition 3.4), and an x0-free reformulation of the forward dynamics (Section 3.4 and Appendix E). They further derive a score-matching training objective and claim that the framework recovers DDBM-VP, DDBM-VE, GOUB, and UniDB as special cases (Proposition 4.1). Experimental results on deraining, super-resolution, and inpainting show improved quantitative metrics over the cited baselines. The paper is carefully written, and the main algebraic derivations of the soft h-function and the forward marginals are internally consistent under the stated regularity conditions.","tokens_in":34281,"tokens_out":4806,"duration_ms":57167,"significance":"If the full chain from the prescribed terminal marginal to the trained sampler were valid, the paper would make a genuine contribution: it gives a clean probabilistic mechanism for avoiding terminal-boundary singularities, provides explicit closed-form objects for a soft bridge, offers a new interpretation of UniDB, and reports strong empirical results on three restoration tasks. The theoretical sections are largely self-contained, and the unification of existing bridge models via parameter identification is appealing. The main value of the paper lies in replacing an endpoint Dirac constraint by a calibrated Gaussian terminal reweighting, which is a natural and potentially useful idea. However, the link between the continuous-time forward SDE and the discrete-time training/sampling procedure contains an acknowledged gap: the reformulated dynamics are only marginally consistent, not pathwise identical to the soft bridge, and the training posterior is constructed from an assumed one-step transition rather than derived from the SDE.","major_comments":[{"comment":"The reverse-time training objective is derived from a posterior p_h(x_{t-1} | x_0, x_t, x_T) that is computed using a 'Gaussian one-step forward transition' in Eq. (74) which is constructed to be consistent with the time marginals of Eq. (73), not derived as the true transition density of the SDE in Eq. (38). The paper itself states in Appendix E.3 that Eq. (38) 'should be interpreted as a marginally consistent reformulation ... rather than as a pathwise identity.' This matters because, for a Gaussian Markov process, matching one-time marginals does not determine the transition law, and the reverse SDE in Eq. (18) requires the conditional score ∇ log p_h(x_t | x_T) of the actual forward process. As written, the learned score may be the score of a different process than the one whose terminal marginal was prescribed. The authors should either prove that the transition in Eq. (74) coincides with the transition of Eq. (38), or explicitly reformulate the method as a discrete-time Markov bridge whose forward transition is Eq. (74) and whose reverse sampler is derived for that discrete process.","section":"Appendix E.3 / A.6"},{"comment":"In the sampling procedure the paper substitutes x_T by (b_T + c_T)x* and trains and evaluates the score network with x* in place of x_T, dropping the source-dependent and noise terms. This substitution is not derived from the forward soft bridge, whose terminal state is random with law N(x*, σ²I). Since the reported results are obtained with this approximation, the consistency between the trained reverse process and the forward soft bridge is broken at a point that is load-bearing for the empirical claims. The authors should state precisely what approximation is being made, provide a formal justification (for example, an argument that the omitted terms are negligible in the relevant limit), or present the reverse process as an approximate sampler whose error is controlled.","section":"Appendix I"},{"comment":"The empirical correction factor ζ is fitted per task to validation performance and is used to rescale the score-driven drift in Eq. (90). Table 4 shows that without ζ the SDE sampler on super-resolution achieves PSNR 22.81, far below the reported SDE-ζ value of 27.22, so the headline SDE results depend critically on this fitted rescaling. The paper calls ζ a discretization correction, but it is not derived from any discretization error analysis. At minimum, the authors should report the sensitivity of all reported metrics to ζ, justify why ζ is needed only for the SDE and not for the ODE in some tasks, and temper the claim that the method is 'parameter-free' in its regularization.","section":"Appendix I / Table 4"},{"comment":"The claim that DDBM-VP, DDBM-VE, and GOUB are recovered as special cases uses the hyperparameter σ̂² = 0, but Theorem 3.2 requires 0 < σ² < σ̄²_{0:T}, which implies σ̂² > 0. Thus σ̂² = 0 lies outside the admissible domain for which the Gaussian terminal marginal is prescribed, and the recovery of hard bridges can only hold in a limiting sense as σ̂² → 0. The proposition should be restated as a limiting recovery, or the admissible domain of the parameters should be extended with a stated limiting argument.","section":"Proposition 4.1"}],"minor_comments":[{"comment":"The text says performance degrades as α approaches the boundary -e^{-θ̄_{0:T}}σ̂²/σ̄²_{0:T}, but the horizontal axis of Figure 2(b) shows only small positive α values; the figure should either include the negative side of the admissible range or the text should clarify what range is shown.","section":"Figure 2(b)"},{"comment":"The notation p_h(x_t, t | x*, T) in the reverse SDE is used before the soft h-conditioned density is formally defined; adding a definition or a pointer would improve readability.","section":"Section 2.2"},{"comment":"The proof of the GOU transition variance uses g_t²/(2θ_t) for the limiting covariance, but this is only valid under the parametrization g_t² = 2λ²θ_t; the notation should state this assumption before using it.","section":"Appendix H"},{"comment":"There are several typographical errors, such as 'right hard' in Appendix A.6 and 'of of' in Appendix E.3, and some figure references (e.g., Figure 3 versus Figure 7/8) are inconsistent. A careful proofreading pass is needed.","section":"Appendix A.6 / E.3"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a useful idea and a considerable amount of correct algebraic work, but the gap between the continuous-time soft bridge and the actual training/sampling procedure is central, and the empirical claims rely on heuristics (ζ and the x_T substitution) that are not derived from the theory. This is fixable within the manuscript's scope if the authors either prove the missing transition-law identity or reframe the method as a discrete-time Markov bridge with an explicitly approximate reverse sampler. I would not reject at this stage, but the revision must address the reverse-process consistency issue head-on."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core theoretical contribution is real and worth your time: prescribing a non-degenerate Gaussian terminal marginal under the h-transform yields a closed-form soft h-function, closed-form forward marginals, and a drift correction that stays finite at T. The unification of DDBM, GOUB, and UniDB by explicit parameter identification (σ̂² = κ⁻¹) checks out and gives UniDB a probabilistic interpretation it was missing. That is new, and it is done carefully.\n\nThe soft spot is in the x0-free reformulation, and it is load-bearing. Appendix E.3 says plainly that Eq. (38) is a marginally consistent reformulation, not a pathwise identity. For a Gaussian Markov process, matching the one-time marginals does not determine the transition law. The reverse SDE in Eq. (18) requires the conditional score of the actual forward bridge, but Section A.6 does not compute that score from Eq. (15); it constructs a Gaussian one-step transition consistent with the marginals and derives the training posterior from that assumed transition. The stress-test note is correct: the score being learned may be the score of a different process. The paper is honest about the concession, but that does not close the gap.\n\nThe empirical section has a separate, smaller problem. Table 4 shows that without the per-task fitted ζ, the SDE sampler on super-resolution is poor (PSNR 22.81 vs 27.22 with ζ). ζ is chosen by validation performance, and there is no code and no error bars. So the state-of-the-art claims are conditional, not demonstrated. This does not refute the math, but it means the reported numbers should be read with caution.\n\nMinor issues: a few typos in the variance algebra in A.6, and the terminal initialization x_T ≈ (b_T + c_T)x⋆ drops the x0 and noise terms without justification. These are fixable.\n\nWho is this for? Anyone working on diffusion bridges or image restoration with bridge models. The theoretical construction is the kind of thing you might cite even if you are skeptical of the empirical claims. It deserves a serious referee. It should not be desk-rejected. My recommendation: send it out, but the referee should push on the x0-free gap. If the authors can either prove the reverse SDE is the true reverse of the prescribed soft bridge, or explicitly scope the x0-free version as a new model whose path measure is not the prescribed one, the paper would be much stronger.\n\nNet: theory solid, gap real, empirics conditional. Worth engaging.","headline":"Solid closed-form soft-bridge theory with a real gap between the marginal-only x0-free reformulation and the reverse SDE used in practice; deserves review, but performance claims need an asterisk.","tokens_in":34718,"tokens_out":1851,"would_cite":true,"duration_ms":21649,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60J60","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Soft Gaussian endpoints remove the terminal singularity of diffusion bridges","keywords":["diffusion bridges","Doob h-transform","terminal singularities","soft terminal constraints","Gaussian terminal marginal","image restoration","conditional score matching","stochastic optimal control"],"falsifier":"Run the trained SDDBM reverse sampler, record the empirical terminal samples it produces, and compare their covariance to the prescribed $\\sigma^2 I$; if the covariance is far from $\\sigma^2 I$, or if shrinking $\\sigma$ collapses the terminal law while the drift stays finite, the sampler is not realizing the soft bridge. A second check is to retrain with the exact $x_0$-conditioned soft bridge and compare against the $x_0$-free variant that starts at $(b_T+c_T)x_\\star$; a sharp quality drop would show that the approximation, not the soft terminal constraint, carries the reported gains.","tokens_in":33686,"feed_emoji":"🖼️","tokens_out":10734,"duration_ms":96044,"temperature":0.7,"pith_summary":"Diffusion bridge models that condition on an exact endpoint pay a structural price: the terminal law collapses to a Dirac measure and the drift coefficients diverge as $t\\to T$. This paper establishes that the cure is to prescribe a non-degenerate Gaussian terminal marginal $\\mathcal{N}(x^\\star,\\sigma^2 I)$ under the Doob $h$-transformed measure, which yields a closed-form soft $h$-function, closed-form forward marginals, and an $x_0$-free reverse dynamics whose drift denominators never vanish. Because $\\widehat{\\sigma}^2>0$ keeps the terminal drift finite, sampling can start at $t=T$ and the noise scale does not degrade at the boundary. The same construction recovers DDBMs, GOUB, and UniDB as parameter limits, and the reported experiments on deraining, super-resolution, and inpainting improve over the compared baselines.","feed_headline":"Soft endpoints stop diffusion bridges from blowing up","feed_subtitle":"A Gaussian target instead of an exact endpoint removes terminal singularities and improves image-restoration quality.","key_machinery":"The central object is the soft $h$-function obtained by Gaussian terminal reweighting: for the GOU reference bridge with transition kernel $p(z,T|x_t,t)=\\mathcal{N}(z;\\bar{\\mu}_{t:T},\\bar{\\sigma}^2_{t:T}I)$ and calibrated terminal weight $q(z)=\\mathcal{N}(z;\\widehat{x},\\widehat{\\sigma}^2 I)$, the function $h(x_t,t)=\\int p(z,T|x_t,t)q(z)\\,dz$ is available in closed form, $h\\propto\\exp(-\\|\\bar{\\mu}_{t:T}-\\widehat{x}\\|^2/[2(\\widehat{\\sigma}^2+\\bar{\\sigma}^2_{t:T})])$. Its gradient adds a drift correction whose denominator $\\widehat{\\sigma}^2+\\bar{\\sigma}^2_{t:T}$ stays strictly positive at $t=T$ precisely because $\\widehat{\\sigma}^2>0$, so the terminal drift no longer blows up. Theorem 3.2 fixes $\\widehat{x}$ and $\\widehat{\\sigma}^2$ uniquely from the prescribed terminal marginal $\\mathcal{N}(x^\\star,\\sigma^2 I)$, yielding closed-form Gaussian forward marginals and a well-posed $x_0$-free reformulation; the admissibility condition $\\phi_t\\neq0$ (equivalently the relaxation parameter $\\rho>0$) delineates the usable parameter regime.","core_discovery":"SDDBMs claim that a finite-coefficient, $x_0$-free affine bridge cannot both forget its source and collapse to a point: its terminal law is necessarily a non-degenerate Gaussian whose mean still depends on $x_0$, so hard endpoint conditioning can only be realized through drift blow-up. The paper therefore prescribes the terminal marginal under the $h$-transformed measure to be $\\mathcal{N}(x^\\star,\\sigma^2 I)$ (with a soft center $x^*$ that may differ slightly from the desired target $x_\\star$), proves that the unique Gaussian terminal weight realizing it is $q=\\mathcal{N}(\\widehat{x},\\widehat{\\sigma}^2 I)$ with the calibration in Eq. (12), and obtains a closed-form soft $h$-function, closed-form forward marginals, and an $x_0$-free reverse SDE with non-vanishing terminal noise. Under specific hyper-parameter choices the framework recovers DDBM-VP, DDBM-VE, GOUB, and UniDB; in particular UniDB's terminal penalty $\\kappa$ becomes an explicit terminal variance $\\sigma^2=\\bar{\\sigma}^2_{0:T}/(\\kappa\\bar{\\sigma}^2_{0:T}+1)$. On deraining, super-resolution, and inpainting, the reported PSNR/SSIM/LPIPS/FID improve over the compared baselines.","pith_inferences":["The same soft-terminal construction should transfer to Schrodinger-bridge or flow-matching transports that currently rely on exact endpoint coupling, with $\\sigma$ acting as a tunable dial between target fidelity and sampling stability.","The paper's use of $x_\\star$ in place of $x_T$ for the score network suggests a broader design principle: condition score networks on a clean target estimate rather than a noisy terminal state, which may extend beyond restoration to any paired translation task.","The empirical correction factor $\\zeta$ hints that the Euler-discretized reverse drift is systematically biased; switching to a higher-order exponential integrator might remove the need for $\\zeta$ and further improve the ODE results."],"forward_implications":["Sampling can start exactly at $t=T$: the drift denominator $\\widehat{\\sigma}^2+\\bar{\\sigma}^2_{t:T}$ stays positive, so no $\\epsilon$-truncation of the terminal interval is needed.","The terminal noise scale $\\bar{\\sigma}'^2_{0:T}$ remains positive, so the standard noise-prediction training objective applies uniformly without boundary-specific score parameterizations.","Tuning $\\sigma$ sweeps a continuous family from a near-hard endpoint ($\\sigma\\to0$) to a weakly guided bridge, making the fidelity-versus-stability trade-off explicit.","The framework contains DDBMs, GOUB, and UniDB as special hyper-parameter cases, so analysis and improvements transfer to those models.","On the reported deraining, super-resolution, and inpainting benchmarks, SDDBMs beat the listed bridge and restoration baselines on PSNR, SSIM, LPIPS, and FID."],"supporting_citations":[{"why":"Defines DDBMs, the hard-endpoint Doob h-transform bridge with score-matching training that SDDBMs generalize and compare against.","marker":"(Zhou et al., 2024)"},{"why":"Supplies the GOU bridge dynamics whose singular denominator motivates the soft terminal constraint and serves as a baseline.","marker":"(Yue et al., 2024)"},{"why":"Introduces UniDB's finite terminal penalty, which SDDBMs recover as a special case and reinterpret probabilistically.","marker":"(Zhu et al., 2025)"},{"why":"Provides the Doob h-transform and GOU transition-density formulas used for the closed-form soft bridge.","marker":"(Särkkä & Solin, 2019)"},{"why":"Gives the reverse-time SDE and conditional score-matching framework that the x0-free reverse dynamics and training objective build on.","marker":"(Song et al., 2021b)"},{"why":"Establishes the image-restoration setup, U-Net architecture, noise schedule, and L1 objective shared by the experiments.","marker":"(Luo et al., 2023)"}],"fun_headline_variants":["Soft terminal constraints tame diffusion bridges","Prescribed Gaussian terminal marginals fix bridge singularities","Soft bridges: from Dirac to Gaussian terminal laws","Diffusion bridges with soft endpoints avoid blow-up","SDDBMs: regularizing bridge terminals with Gaussian noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the unproved approximation that the $x_0$-free reverse process—initialized at $x_T\\approx(b_T+c_T)x_\\star$, with the score network fed $x_\\star$ and the drift rescaled by a per-task $\\zeta$—remains faithful to the forward soft bridge; if that approximation fails, the mathematical clean-up and the reported gains come apart.","fun_headline_variants_meta":{"raw":{"variants":["Soft terminal constraints tame diffusion bridges","Prescribed Gaussian terminal marginals fix bridge singularities","Soft bridges: from Dirac to Gaussian terminal laws","Diffusion bridges with soft endpoints avoid blow-up","SDDBMs: regularizing bridge terminals with Gaussian noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000594,"raw_usage":{"total_tokens":2840,"prompt_tokens":1063,"completion_tokens":1777,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":1706}},"tokens_in":679,"tokens_out":1777,"duration_ms":13418,"temperature":1.0,"reasoning_tokens":1706,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:30:34.240342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the trained SDDBM reverse sampler, record the empirical terminal samples it produces, and compare their covariance to the prescribed $\\sigma^2 I$; if the covariance is far from $\\sigma^2 I$, or if shrinking $\\sigma$ collapses the terminal law while the drift stays finite, the sampler is not realizing the soft bridge. A second check is to retrain with the exact $x_0$-conditioned soft bridge and compare against the $x_0$-free variant that starts at $(b_T+c_T)x_\\star$; a sharp quality drop would show that the approximation, not the soft terminal constraint, carries the reported gains.","supporting_citations":[],"review_version":1}