{"id":"70f48841-2a9a-4232-9315-aeab3387530d","arxiv_id":"2509.03910","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A single invertible map constructed from two triangular normalizing flows can conditionally sample both the likelihood and the posterior in Bayesian inverse problems.","lead":"The paper builds a single invertible mathematical map that can both simulate data from a model and infer hidden causes from measurements, unifying two tasks usually done by separate neural networks. It works by gluing together two 'triangular' maps so that running the map one way produces forward samples and running it backward produces posterior samples.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Framework requires invertible conditional maps; fails for deterministic likelihoods despite claimed handling of singular distributions.","rationale":"The reader's weakest_assumption correctly identifies the reliance on invertible triangular maps and the failure for degenerate likelihoods. This is indeed the most load-bearing concern because the paper's headline contribution—a single invertible map for both simulation and inference—is only guaranteed under non-atomic conditional distributions. The mathematical core is otherwise sound: given the existence of such maps, the proofs of Lemmas 3.3, 3.4 and Theorem 3.7 are correct, and the construction works for the Gaussian and non-linear examples with full-rank noise. The issue is a scope limitation, not an internal contradiction. The paper explicitly gestures at singular distributions in Section 3.1, but the measure-theoretic language does not rescue the construction; invertibility is a genuine requirement. Therefore the verdict remains CONDITIONAL: the paper should clearly state the non-degeneracy assumption (e.g., atomless conditionals) and temper the claim about singular data distributions. No change to the reader's conditional verdict is needed, but the concern should be highlighted in revisions.","tokens_in":12412,"tokens_out":9214,"duration_ms":89373,"concrete_test":"Analytical check: Let U~N(0,1) and F=U. Assume there exists an invertible lower-triangular qF(x,y)=(q1(x), q2(x,y)) with qF#N(0,I)=μU,F. For each x, the slice y↦q2(x,y) must push N(0,1) forward to δ_{q1(x)}, forcing q2(x,·) to be constant a.e., contradicting invertibility. Equivalently, compute Flike(y;u)=qF2(qF1^{-1}(u),y) from (4); it is independent of y, so S is not injective and Theorem 3.7's S^{-1}=R fails. This settles whether the framework extends to deterministic likelihoods.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central construction (Theorem 3.7) assumes the existence of invertible lower- and upper-triangular maps qF and pF such that qF#μX,Y = pF#μX,Y = μU,F. The proof and the definition of S in (4)-(5) require the conditional maps Flike(y;u) and Fpost(x;f) to be invertible in y and x, respectively. This fails whenever a conditional distribution has atoms. In particular, for a deterministic forward model f=K(u), the likelihood μF|U=u is a point mass, so Flike(y;u) is constant in y and not invertible; S is then not a bijection and S^{-1}=R cannot hold. Section 3.1 explicitly claims to handle 'singular data distributions (e.g., data concentrated on manifolds)', but the proofs of Lemmas 3.3, 3.4 and Theorem 3.7 rely on the inverses qG2, pG1, and pF2^{-1}, which do not exist for atomic conditionals. The Gaussian example notes that qF degenerates as ΣF→0, but only for square invertible K does the limiting S remain invertible; for underdetermined problems (m>n) the limiting S is rank-deficient. Thus the framework excludes a central class of noiseless inverse problems, and the paper overstates its scope by suggesting singular distributions are handled.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a single invertible map S : R^{m+n} -> R^{m+n} for Bayesian inverse problems. Given an invertible lower-triangular map qF and an invertible upper-triangular map pF that both push a product reference measure μX ⊗ μY to the joint μU,F, the authors define S(u,y) = (F_post^{-1}(u; F_like(y;u)), F_like(y;u)) and R = S^{-1}. Theorem 3.7 shows that R_1(·,f) and S_2(u,·) generate the posterior and likelihood, respectively, and that R and S exchange μX,F and μU,Y. Training objectives based on measure distances are given in Corollary 3.11, with consistency conditions in Theorem 3.13. Numerical illustrations include a Gaussian linear example, a nonlinear toy problem, and MNIST inpainting.","tokens_in":12759,"tokens_out":8795,"duration_ms":84471,"significance":"If the main theorem holds, the paper gives a clean and elegant unification of amortized posterior inference and likelihood simulation in a single invertible architecture. The proofs are short and essentially correct under the stated invertibility assumptions; the construction is explicit and does not fit constants to make the results true. The Gaussian example provides a concrete check. The main caveat is that the framework requires non-atomic absolutely continuous conditionals with invertible triangular transports, so the claim to handle 'singular data distributions' in Section 3.1 is not correct. With the scope stated honestly, the paper would be a useful conceptual contribution to conditional generative modeling.","major_comments":[{"comment":"The paper states in §3.1 that it formulates results in terms of measures 'to handle singular data distributions (e.g., data concentrated on manifolds).' This is not supported by the construction. Lemma 3.3 and Lemma 3.4 require qF and pF to be invertible triangular maps, and (4)–(5) require F_like(·;u) and F_post(·;f) to be invertible. For a deterministic forward model f = K(u), μF|U=u is a point mass; any lower-triangular qF with qF2(u,·)#μY = δ_{K(u)} would make qF2 independent of y, so ∂qF2/∂y is singular and qF cannot be invertible on R^{m+n}. Hence S is not a bijection and S^{-1}=R fails. The same problem occurs for any atomic conditional. Please either restrict the theorems explicitly to non-atomic conditionals (e.g., absolutely continuous with positive densities) or provide a separate treatment of the degenerate limit; the current claim overstates the scope.","section":"§3.1, Lemmas 3.3–3.4, Theorem 3.7"},{"comment":"The displayed upper-triangular map pF has a typo in the top-right block: it reads ΣU K^T (KΣU K^T + ΣU)^{-1/2}; the last covariance should be ΣF, and the orientation of the matrix square root must be specified consistently. As written, the example does not satisfy pF pF^T = ΣU,F and is not reproducible. This is a localized error, but it is in the main illustrative example and should be fixed.","section":"§4.1, pF matrix above Eq. (6)"},{"comment":"The numerical evidence is qualitative. There are no error bars, no comparisons with existing conditional normalizing flows or direct inversion baselines, and no quantitative metrics for posterior sample quality. The statement in §5 that the construction has 'improved conditioning' is supported only by one toy condition-number plot. If the paper is to claim practical advantages, these need quantitative support; otherwise the claims should be moderated.","section":"§4.2–4.3 and §5"}],"minor_comments":[{"comment":"The claim that the triangular structure 'guarantees' a unique solution to the variational problems in Lemma 3.5 is too strong unless additional monotonicity and parameterization assumptions are imposed. Please qualify or remove.","section":"Remark 3.6"},{"comment":"Typos: 'in to one' should be 'into one'; 'with a a linear' should be 'with a linear'; 'Fischer information' should be 'Fisher information.'","section":"Abstract and §4.1"},{"comment":"The map qF uses the sign function σ, which is not differentiable at 0. The density transformation via |∇qF^{-1}| is then not defined pointwise. Please clarify that the example is intended in a piecewise sense or use a smooth approximation.","section":"§4.2"},{"comment":"Several figures lack axis labels and legends; adding these would improve interpretability, especially for the conditional-distribution and condition-number plots.","section":"Figures 6–8"}],"recommendation":"major_revision","confidential_remarks":"The central derivation is sound under the right assumptions, and the singular-distribution issue is fixable by an honest scope statement plus a discussion of degenerate limits. The author's own prior work [42] is cited only for experimental context and does not feed into the proofs, so there is no circularity concern. The paper fits the journal's scope, but the numerical validation needs strengthening before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the main construction is sound and new. The paper takes two triangular conditional maps—one for the likelihood, one for the posterior—and composes them into a single invertible map S whose inverse gives the other direction. Theorem 3.7 and the loss identities in 3.13 check out given the assumptions. That's genuinely useful. I don't see circularity and the citations look appropriate. The numerical examples are only illustrative (the MNIST one is an affine fit to a Gaussian), so don't treat them as validation.\n\nThe real soft spot is scope. The construction requires the conditional maps to be invertible in the conditioning variable. That means both likelihood and posterior have to be absolutely continuous (or at least atomless with a measurable inverse). Deterministic forward models f=K(u) give a point-mass likelihood, so Flike is constant in y, S isn't a bijection, and Theorem 3.7 doesn't apply. Section 3.1 says the paper handles 'singular data distributions (e.g., data concentrated on manifolds)', but that refers to the joint/marginal measures; the proofs of Lemmas 3.3–3.4 and Theorem 3.7 need the inverses of the conditional maps, which don't exist for atomic conditionals. The Gaussian example even notes qF degenerates as Sigma_F -> 0, but only for square invertible K does S stay invertible in the limit; for underdetermined problems the limiting S is rank-deficient. The stress-test note is right: the paper overstates its scope.\n\nMinor issues: the typo in the Gaussian example (top-right block of pF should have (K Sigma_U K^T + Sigma_F)^{-1/2}, not Sigma_U in the denominator), Remark 3.6 overclaims uniqueness of the variational problem, and there are no error bars, baselines, or code. Those are fixable.\n\nOverall: a solid methodological contribution for Bayesian inverse problems with noisy likelihoods and non-degenerate posteriors. It deserves a serious referee. I'd send it to review with a request to fix the scope statement, the typo, and the uniqueness remark. I'd likely cite it if I were working on amortized UQ; it's a clean unification. For a reading group, it's a maybe—good to walk through the proofs once.","headline":"A genuinely new invertible-map construction for joint likelihood/posterior sampling, correct under absolutely-continuous conditionals but overclaimed for singular/noiseless cases.","tokens_in":13193,"tokens_out":4430,"would_cite":true,"duration_ms":40352,"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 single invertible generative map, built by composing an upper- and a lower-triangular normalizing flow, can serve both as a likelihood simulator and as a posterior sampler for inverse problems.","keywords":["inverse problems","Bayesian inference","normalizing flows","triangular maps","conditional generative models","measure transport","likelihood simulation","posterior sampling"],"falsifier":"In the linear-Gaussian test of Section 4.1, set the measurement noise covariance to zero and train the two triangular flows by maximum likelihood; the likelihood map becomes constant in the latent y, so S in (4) is not injective. Checking the Jacobian determinant of S at two distinct y values, or attempting to evaluate S^{-1} at (x, f) with f = K(u), will show the inverse does not exist, contradicting Theorem 3.7.","tokens_in":12335,"feed_emoji":"🔄","tokens_out":9376,"duration_ms":86376,"temperature":0.7,"pith_summary":"The paper sets out to give inverse problems one generative model that works in both directions: run it forward to simulate measurements from a proposed unknown, run it backward to sample unknowns from the posterior given a measurement. It claims that composing an upper-triangular conditional flow (for the posterior) with a lower-triangular conditional flow (for the likelihood) yields a single invertible map S whose inverse is exactly the map that does the other task. The paper proves the relevant measure-transport identities, proposes a training loss that acts directly on S and S−1, and demonstrates the construction on linear-Gaussian, nonlinear, and image-inpainting examples. A reader should care because this replaces the usual two-model or expensive-MCMC setup with one learned mapping that can be evaluated in both modes.","feed_headline":"One invertible map does both simulation and Bayesian inference","feed_subtitle":"Triangular flows combined into one map S let the same model sample likelihoods forward and posteriors backward.","key_machinery":"The central object is a triangular map: a differentiable bijection whose Jacobian is triangular. Lower-triangular maps have the form F(x,y) = (F1(x), F2(x,y)); upper-triangular maps have the form F(x,y) = (F1(x,y), F2(y)). Because of this structure, conditioning on one coordinate reduces to function evaluation or inversion in the other coordinate, which lets a single joint transport map expose both conditionals. The paper composes these two triangular structures into S and shows the inverse is available in closed form, so the same network architecture can be run forward for simulation and backward for inference. The proposed training loss is a measure distance between (u, S2(u,y)) and the jo","core_discovery":"Formally, the maps qF (lower-triangular) and pF (upper-triangular) each push a product reference measure μX⊗μY onto the joint distribution μU,F. From them the paper constructs S = (S1, S2) and R = (R1, R2) by (4)-(5): S1(u,y) = Fpost^{-1}(u; Flike(y;u)), S2(u,y) = Flike(y;u), R1(x,f) = Fpost(x;f), R2(x,f) = Flike^{-1}(f; Fpost(x;f)). Theorem 3.7 shows S^{-1} = R and the four push-forward identities: R1(·,f)#μX = μU|F=f, S2(u,·)#μY = μF|U=u, R#μX,F = μU,Y, and S#μU,Y = μX,F. The claim is exact, not approximate: if the two triangular maps represent the joint distribution, then forward evaluation of S samples the likelihood and forward evaluation of R samples the posterior; moreover S pushes μU","pith_inferences":["Editorial inference: the framework is most naturally extended to non-deterministic generators by absorbing the stochasticity into the latent variable, which may let diffusion-type models inherit the same forward/backward duality.","Editorial inference: the noiseless-limit failure suggests testing a relaxed variant that replaces the point-mass likelihood with a small-noise approximation; the conditioning behavior in Section 4.1 indicates S may still provide useful posterior samples as the noise goes to zero even when exact invertibility degenerates.","Editorial inference: one can use Theorem 3.13 as a model-selection test: train with J1 only, then measure J4(R); if J4 fails, the learned S does not actually transport the joint distribution, remedying a gap that purely conditional losses might miss.","Editorial inference: the same triangular composition could be applied to condition on any fixed subset of variables, not just the unknown/measurement split, giving a general recipe for conditional generative modelling from joint samples."],"forward_implications":["A single trained map replaces separate likelihood-simulator and posterior-sampler models in any Bayesian inverse problem whose joint distribution admits triangular transports.","Inference becomes one function evaluation of S^{-1}, so posterior samples are available at the cost of a forward pass, without iterative solvers or MCMC burn-in.","The construction gives exact conditional sampling whenever the triangular flows are trained to zero loss, and the paper's conditioning analysis suggests S can be better conditioned than either triangular flow alone.","The consistency identities give a practical diagnostic: check the joint pushforward to certify the conditional samplers.","Because the map is explicitly invertible, the same model can be reused for experimental design by amortizing over a design parameter."],"supporting_citations":[{"why":"The invertible-network approach to inverse problems that this construction builds on and contrasts with; supplies the use case of one network for posterior sampling.","marker":"[5]"},{"why":"Supplies the triangular transport-map lemma that the paper reproduces to obtain conditional sampling from triangular maps.","marker":"[27]"},{"why":"Amortized Bayesian inference with invertible networks; the baseline posterior-sampling approach whose two-map limitation this work addresses.","marker":"[35]"},{"why":"Jointly amortized neural approximation of Bayesian models; another baseline that trains simulation and inference separately.","marker":"[36]"},{"why":"Earlier experiments with affine maps that include the forward operator, used here as the practical parametrization for S.","marker":"[42]"},{"why":"Triangular-map parametrization used to compute the nonlinear example's flows and conditionals.","marker":"[31]"}],"fun_headline_variants":["One invertible map: sample likelihoods and posteriors","Same map does simulation and Bayesian inference exactly","Two triangular flows fused into one invertible generator","Forward and inverse problems solved by a single map"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The construction assumes the joint distribution of unknowns and measurements can be reached by two invertible triangular transformations from independent noise, and that the resulting conditional transformations can be inverted; if the measurements are a deterministic function of the unknown (no noise), the likelihood is a single point, the conditional transformation cannot be inverted, and the single-map construction breaks down.","fun_headline_variants_meta":{"raw":{"variants":["One invertible map: sample likelihoods and posteriors","Same map does simulation and Bayesian inference exactly","Two triangular flows fused into one invertible generator","Forward and inverse problems solved by a single map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000155,"raw_usage":{"total_tokens":1042,"prompt_tokens":728,"completion_tokens":314,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":254}},"tokens_in":472,"tokens_out":314,"duration_ms":3743,"temperature":1.0,"reasoning_tokens":254,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:33:09.395055+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the linear-Gaussian test of Section 4.1, set the measurement noise covariance to zero and train the two triangular flows by maximum likelihood; the likelihood map becomes constant in the latent y, so S in (4) is not injective. Checking the Jacobian determinant of S at two distinct y values, or attempting to evaluate S^{-1} at (x, f) with f = K(u), will show the inverse does not exist, contradicting Theorem 3.7.","supporting_citations":[{"cited_title":"Analyzing inverse problems with invertible neural networks","cited_arxiv_id":null,"evidence_quote":"The invertible-network approach to inverse problems that this construction builds on and contrasts with; supplies the use case of one network for posterior sampling."},{"cited_title":"Sam- pling via measure transport: An introduction","cited_arxiv_id":null,"evidence_quote":"Supplies the triangular transport-map lemma that the paper reproduces to obtain conditional sampling from triangular maps."},{"cited_title":"Radev, Ulf K","cited_arxiv_id":null,"evidence_quote":"Amortized Bayesian inference with invertible networks; the baseline posterior-sampling approach whose two-map limitation this work addresses."},{"cited_title":"Radev, Marvin Schmitt, Valentin Pratz, Umberto Picchini, Ull- rich K¨ othe, and Paul-Christian B¨ urkner","cited_arxiv_id":null,"evidence_quote":"Jointly amortized neural approximation of Bayesian models; another baseline that trains simulation and inference separately."},{"cited_title":"The reversible simulator – a data-driven approach for solving forward and inverse problems","cited_arxiv_id":null,"evidence_quote":"Earlier experiments with affine maps that include the forward operator, used here as the practical parametrization for S."},{"cited_title":"Monotone Parameterization Toolbkit (MParT),","cited_arxiv_id":null,"evidence_quote":"Triangular-map parametrization used to compute the nonlinear example's flows and conditionals."}],"review_version":1}