{"id":"90a7b5a0-2dcf-45ff-81fe-ec3adde13e96","arxiv_id":"2607.26264","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A symmetrized Sinkhorn divergence, averaging transport costs of a signal and its negation, yields better Gibbs posterior inference for oscillatory inverse problems.","lead":"The paper introduces a sign-symmetric optimal-transport loss for comparing oscillatory signals and embeds it in a Gibbs posterior for inverse problems. It claims smoother risk landscapes and more reliable uncertainty estimates than Euclidean or trace-wise Wasserstein losses.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Robustness theorems rely on an unproved H^{-1}-Lipschitz step and on density lower bounds that softplus does not guarantee; without these, Thms. 2-3 do not support the claimed noise robustness.","rationale":"The reader's conditional verdict correctly identifies that the robustness theorems require positive density lower bounds that the softplus construction does not guarantee. I agree with that, but there is an even more direct gap: Proposition 4's second inequality does not follow from L-infinity-Lipschitzness of the normalization operator. This is the step that lets Theorems 2-3 conclude posterior stability in H^{-1}; if it fails, the theoretical robustness guarantee is unproved even when densities are well separated from zero. The numerical experiments are suggestive and the benchmark is designed to isolate cycle skipping, so I would not reject the paper on this basis. However, Contribution 3 and the abstract explicitly advertise derived robustness guarantees; a corrected proof or a clear restriction of the theorem assumptions is needed. The proposed numerical test would settle whether the Lipschitz inequality is salvageable without additional smoothness assumptions, and whether the density lower bound is actually satisfied in the reported experimental regime.","tokens_in":27691,"tokens_out":18530,"duration_ms":187809,"concrete_test":"Verify Proposition 4 numerically on X=[0,1]: take f as a zero-mean L-infinity step function and f' = f + epsilon sin(k pi x); compute R_k = ||T_sigma_delta f - T_sigma_delta f'||_{H^{-1}} / ||f - f'||_{H^{-1}} for increasing k and fixed epsilon, delta. If R_k is unbounded as k grows, the M2 inequality is false without extra assumptions. Also compute min_x T_sigma_delta g and min_x T_sigma_delta f over the benchmark grids; if the minimum is below, say, 10^{-3} of the maximum, the positive-lower-bound assumption in Proposition 5 is violated, so the robustness constants are uncontrolled for the reported experiments.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 4 claims ||Tsigma f - Tsigma f'||_{H^{-1}} <= L_sigma ||f - f'||_{H^{-1}}, justified by Tsigma being locally Lipschitz in L-infinity and by the dual definition of H^{-1}. This is a non sequitur: L-infinity-Lipschitzness controls L1 pairings, hence gives bounds in terms of ||f - f'||_{L1}, not the weaker H^{-1} norm. Without additional regularity of f, multiplication by sigma'(f) is not known to be bounded on H^{-1}; Proposition 5 and Corollary 2 inherit this gap, and Theorems 2-3 use exactly these estimates. Separately, Proposition 5 requires the normalized densities T_sigma f, T_sigma g, T_sigma g' to be bounded below by a positive constant. For softplus sigma_delta(z)=log(1+e^{delta z}), the normalized density is ~ e^{delta f}/Z on negative regions, so it can be arbitrarily close to zero; the W2-to-H^{-1} equivalence constant [44] is then uncontrolled. Since the abstract's 'greater robustness to observational noise' is partly justified by these theorems, the theoretical pillar of the central claim is conditional: it needs either a corrected proof under appropriate regularity/lower bounds or a restriction of the robustness claims to regimes where both conditions hold.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a Gibbs-posterior inference framework for oscillatory inverse problems, built on a symmetrized Sinkhorn divergence. Signed signals are mean-centered, mapped to probability densities via a softplus-type normalization, and compared through the sum of Sinkhorn divergences of the normalized signals and their negations. The authors claim smoothness, asymptotic convexity, well-definedness of the Gibbs posterior, and H^{-1}-Lipschitz robustness to data and forward-model perturbations. Numerical experiments on a synthetic seismic benchmark compare the proposed loss with Euclidean and trace-wise Wasserstein losses, reporting smoother risk landscapes, more accurate posterior means, and better population-level recovery.","tokens_in":28062,"tokens_out":3861,"duration_ms":43353,"significance":"If the theory and experiments hold, the framework would be a useful contribution: it is forward-model-agnostic, provides a principled way to apply transport-based divergences to signed oscillatory data, and includes an adaptive MCMC scheme that learns the inverse temperature. The numerical study is carefully structured, with single-event, noise-robustness, and population-level experiments, and the results are consistent with the claimed qualitative advantages. However, the theoretical robustness results are presently conditional on assumptions that are neither verified for the chosen normalization nor shown to hold for the benchmark signals. The paper does not provide code or data, and the benchmark is synthetic. The central contribution is therefore promising but not yet fully supported at the level claimed in the abstract.","major_comments":[{"comment":"The proof of the second inequality asserts that Tσ is locally Lipschitz in L∞ and then concludes a Lipschitz bound in Ḣ^{-1}. This does not follow: Lipschitz continuity in L∞ controls pairings with L1 test functions, and hence gives a bound in terms of ||f−f'||_{L1}, not the weaker Ḣ^{-1} norm. Multiplication by σ'(f) is not in general a bounded operator on Ḣ^{-1} without additional regularity on f. Since Proposition 5, Corollary 2, and Theorems 2–3 all rely on this step, the robustness estimates are not established as written.","section":"Section 4.1, Proposition 4"},{"comment":"The assumption that the normalized densities Tσf, Tσg, Tσg' are bounded below by a positive constant is not guaranteed for the softplus function σδ(z)=log(1+e^{δz}) used throughout the experiments. For signals taking sufficiently negative values on parts of the domain, Tσf is exponentially small in δ and can be arbitrarily close to zero. The W2-to-Ḣ^{-1} equivalence from [44] requires such lower bounds, and the constant Cε in Proposition 5 depends on them. The manuscript should either impose and verify a quantitative lower-bound condition for the chosen normalization, or restrict the robustness claims to regimes where that condition provably holds for the benchmark signals.","section":"Section 4.3, Proposition 5"},{"comment":"The Hellinger robustness estimates are presented as unconditional guarantees, but they inherit the gaps in Proposition 4 and Proposition 5. In particular, Theorem 3 applies Corollary 2 with the first argument perturbed; this is formally justified by symmetry of D^2, but it still depends on the same unproved Ḣ^{-1} Lipschitz step. The manuscript should explicitly state the regularity and density lower-bound conditions as hypotheses of Theorems 2–3 and discuss whether they are satisfied by typical oscillatory signals. Otherwise the abstract's claim of 'greater robustness to observational noise' is supported only empirically.","section":"Theorems 2–3 and Corollary 2"},{"comment":"The convexity results are asymptotic as δ→∞, and Remark 3 correctly concedes that uniform convexity may fail for finite δ. This is not a logical error, but it weakens the theoretical support for the claimed 'favorable convexity behavior' in the abstract and Contribution 3. The manuscript should more clearly state that finite-δ convexity is not established and that the reduced spurious-minima behavior rests on numerical evidence.","section":"Propositions 2–3 and Remark 3"}],"minor_comments":[{"comment":"The paper states that code and data are available from the corresponding author upon request and will be made public upon acceptance. For a numerical paper whose central claims are empirical, a permanent repository link at submission time would materially aid reproducibility.","section":"Data availability"},{"comment":"The benchmark is synthetic and designed by the authors, and no comparison is made with a fully Bayesian likelihood-based posterior. This should be acknowledged as a limitation in Section 5.1 or the conclusions.","section":"Section 5.1"},{"comment":"The discrete loss is denoted Sσ and the continuous loss Sσ,ε; the dependence on ε in the discrete setting is implicit (appearing in the Sinkhorn divergence definition). A brief remark would help avoid confusion.","section":"Notation, Eq. (10) vs. Eq. (17)"},{"comment":"The pointwise limit of σδ(f) for f<0 is stated as 0, which is correct, but the convergence is not uniform near f=0. This nonuniformity is the source of the Remark 3 caveat and deserves a sentence in the proof rather than only in the remark.","section":"Section 4.1, proof of Proposition 2"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the unproved Ḣ^{-1} Lipschitz property in Proposition 4 and the unverified density lower-bound assumption in Proposition 5. These are load-bearing for Theorems 2–3 and for the robustness claim in the abstract. The paper could be acceptable after either a corrected proof under explicit regularity/lower-bound assumptions, or a clear weakening of the theoretical robustness claims so that the numerical evidence carries the weight. I would also encourage the authors to deposit code before a second round."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know: the symmetrized Sinkhorn-Gibbs loss is a plausible and useful extension for oscillatory inverse problems, but the paper's robustness theorems (Thms 2–3) are not actually proven. Proposition 4's H^{-1} Lipschitz step is a non sequitur, and the lower-bound assumption on normalized densities is not guaranteed. So the theoretical grounding is conditional, though the empirical results are encouraging.\n\nWhat's new: the paper sums the Sinkhorn divergence between normalized signals and their normalized negations. That's a simple, sensible way to retain both positive and negative phase information when transport requires positive densities. It extends the trace-wise Wasserstein Bayesian inversion line (Motamed & Appelö, Dunlop & Yang) to full space-time Sinkhorn with a symmetrized loss. The numerical benchmark is carefully constructed: risk landscapes, posterior mean and variance across resolutions, noise robustness measured by Hellinger distance, and population-level recovery with a reasonable oracle reference. Improvements over Euclidean and trace-Wasserstein losses are consistent.\n\nWhere it's soft: the L∞-Lipschitzness of the softplus normalization maps T_σ does not imply Lipschitz continuity in the H^{-1} norm. L∞ control gives you L1 or L∞ bounds on differences; the H^{-1} norm is weaker, and multiplication by σ'(f) is not known to be bounded on H^{-1}. The proof of Proposition 4 asserts this directly, and Proposition 5 and Corollary 2 inherit the gap. On top of that, Prop. 5 assumes normalized densities are bounded below by a positive constant; softplus doesn't guarantee that — for negative excursions σ_δ(f) ≈ e^{δ f}, which can be arbitrarily small, and the W2-to-H^{-1} equivalence constant [44] is then uncontrolled. The paper also concedes in Remark 3 that convexity only holds as δ→∞, not for the finite δ used in experiments. So the theoretical pillar is conditional: either the proofs need correction under appropriate regularity/lower-bound assumptions, or the robustness claims should be restricted to regimes where both conditions hold. The empirical claims are not invalidated by this, but they stand without solid theoretical backing. Also, no code or data are currently public; the data availability statement says available “upon reasonable request.”\n\nWho it's for: people working on transport-based inverse problems and seismic UQ. The idea deserves a serious referee, but I would not lean on Thms 2–3 as they are. I'd send it to review with a clear request to fix the H^{-1} argument and make the assumptions explicit.","headline":"The symmetrized Sinkhorn construction is a genuinely useful empirical idea, but the robustness theorems rely on an unproved H^{-1} Lipschitz step, so the paper's theoretical claims are conditional; worth peer review but needs a fix.","tokens_in":28502,"tokens_out":3405,"would_cite":false,"duration_ms":34008,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49Q22","62F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that comparing a signed signal and its negation through a symmetrized Sinkhorn divergence gives Gibbs posteriors with smoother risk landscapes, stronger noise robustness, and better population recovery than Euclidean or tra","keywords":["oscillatory inverse problems","Gibbs posterior","Sinkhorn divergence","optimal transport","cycle skipping","uncertainty quantification","seismic inversion","signed signal normalization"],"falsifier":"Take a one-dimensional oscillatory signal with a deep negative trough such that δ times the maximum negative amplitude is large, compute the softplus-normalized density on a fine grid, and check whether its minimum value falls below, say, 1e-8; if so, the positive-lower-bound assumption of Proposition 5 fails, and a direct numerical test of the claimed Hellinger-in-H^{-1} bound should show the linear robustness estimate is violated for suitable small perturbations.","tokens_in":27616,"feed_emoji":"🌊","tokens_out":4654,"duration_ms":45222,"temperature":0.7,"pith_summary":"The paper tries to establish that for oscillatory inverse problems, where pointwise losses suffer from cycle skipping and spurious local minima, a Gibbs posterior built on a symmetrized Sinkhorn divergence—comparing a softplus-normalized signal and its normalized negation—produces more reliable uncertainty quantification than Euclidean or trace-wise Wasserstein losses. It proves that the loss is smooth and positive definite, that the resulting Gibbs posterior is well defined, and that the posterior is robust to perturbations of both the data and the forward model when those perturbations are measured in the negative Sobolev norm. Numerical experiments on a seismic benchmark show the symmetrized loss yields less oscillatory risk landscapes, more accurate posterior inference, and population-level recovery close to the oracle distribution.","feed_headline":"Symmetrized Sinkhorn loss tames oscillatory inverse problems","feed_subtitle":"A two-sided optimal-transport loss yields Gibbs posteriors that resist cycle skipping and noise in seismic benchmarks.","key_machinery":"The key object is the symmetrized normalized squared Sinkhorn loss D^2_{σ,ε}(f,g) = S^2_{σ,ε}(f,g) + S^2_{σ,ε}(-f,-g), where S_{σ,ε} is the Sinkhorn divergence—entropically regularized optimal transport—applied to softplus-normalized zero-mean functions. The softplus map σ_δ(z)=ln(1+e^{δz}) turns signed oscillations into positive probability densities, and the negation term captures the complementary negative phase. This loss supplies the empirical risk in a Gibbs posterior Π(dθ) ∝ exp(-λ Φ(θ;g)) Π0(dθ), and its smoothness and Lipschitz bounds in the negative Sobolev norm carry the well-definedness and robustness theorems.","core_discovery":"The central claim is that the nonconvexity of oscillatory inverse problems can be tamed by comparing both a signed signal and its negation in transport space. After mean-centering, the paper maps a signal and its negation through a softplus normalization to probability densities; the loss is the sum of the squared Sinkhorn divergences between normalized prediction and data and between normalized negated prediction and negated data. This symmetrization preserves information from both phases of oscillation, restores asymptotic convexity as the softplus sharpness tends to infinity, and makes the Gibbs posterior Lipschitz stable in the negative Sobolev norm. The paper argues that these propertie","pith_inferences":["A natural extension the paper leaves implicit is that symmetrization could transfer to any signed feature data where both phases carry information, such as medical imaging or quantum state population measurements, not just seismic wavefields.","The asymptotic convexity result suggests an adaptive annealing strategy for the softplus sharpness δ during MCMC—starting at moderate δ and increasing it—which the paper does not explore.","The paper's H^{-1} framing indicates the method is best suited to low-frequency or smooth model errors; testing on high-frequency model perturbations would probe the boundary of the stated guarantee.","A testable practical extension is automatic calibration of the Sinkhorn regularization ε and softplus sharpness δ, which are currently set manually in the experiments."],"forward_implications":["If correct, the framework provides a model-agnostic empirical risk that can be coupled with any forward model, since the loss itself does not depend on the PDE or simulator structure.","Posterior accuracy should improve rather than stagnate as data resolution increases, because the symmetrized loss avoids the wrong-mode concentration seen with Euclidean risks.","The H^{-1} robustness bounds imply that low-frequency perturbations of the data or forward model have controlled influence on the posterior in Hellinger distance, which is directly relevant to uncertainty quantification under model error.","Population-level inference from multiple noisy events should track the oracle empirical distribution closely, enabling recovery of the underlying parameter distribution.","The adaptive sampler treats the inverse temperature as an unknown, so the method can be deployed without manual calibration of the learning rate in multi-parameter settings."],"fun_headline_variants":["Two-sided Sinkhorn loss smooths oscillatory inverse problems","Signed-signal pair stabilizes Gibbs posteriors for oscillatory data","Symmetrized Sinkhorn Gibbs reduces cycle skipping","Two-sided transport loss tames seismic cycle-skipping"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The load-bearing premise, flagged by the paper only for asymptotic convexity in Remark 3, is that softplus-normalized densities stay bounded above and below by positive constants, which is not guaranteed for signals with deep negative or positive excursions because normalized mass can approach zero exponentially.","fun_headline_variants_meta":{"raw":{"variants":["Two-sided Sinkhorn loss smooths oscillatory inverse problems","Signed-signal pair stabilizes Gibbs posteriors for oscillatory data","Symmetrized Sinkhorn Gibbs reduces cycle skipping","Two-sided transport loss tames seismic cycle-skipping"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00056,"raw_usage":{"total_tokens":2510,"prompt_tokens":771,"completion_tokens":1739,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":1669}},"tokens_in":515,"tokens_out":1739,"duration_ms":12922,"temperature":1.0,"reasoning_tokens":1669,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T00:19:07.548379+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a one-dimensional oscillatory signal with a deep negative trough such that δ times the maximum negative amplitude is large, compute the softplus-normalized density on a fine grid, and check whether its minimum value falls below, say, 1e-8; if so, the positive-lower-bound assumption of Proposition 5 fails, and a direct numerical test of the claimed Hellinger-in-H^{-1} bound should show the linear robustness estimate is violated for suitable small perturbations.","supporting_citations":[],"review_version":1}