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An invertible generative model for forward and inverse problems

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A genuinely new invertible-map construction for joint likelihood/posterior sampling, correct under absolutely-continuous conditionals but overclaimed for singular/noiseless cases. read the letter →

arxiv 2509.03910 v1 pith:RIPXYMBY submitted 2025-09-04 stat.ML cs.LGmath.PR

Tristan van Leeuwen , Christoph Brune , Marcello Carioni This is my paper
classification stat.MLcs.LGmath.PR
keywords inverseproblemsBayesianinferencenormalizingflowstriangularmapsconditionalgenerativemodelsmeasuretransportlikelihoodsimulationposteriorsampling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§3.1, Lemmas 3.3–3.4, Theorem 3.7] 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.
  2. [§4.1, pF matrix above Eq. (6)] 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.
  3. [§4.2–4.3 and §5] 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.
minor comments (4)
  1. [Remark 3.6] 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.
  2. [Abstract and §4.1] Typos: 'in to one' should be 'into one'; 'with a a linear' should be 'with a linear'; 'Fischer information' should be 'Fisher information.'
  3. [§4.2] 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.
  4. [Figures 6–8] Several figures lack axis labels and legends; adding these would improve interpretability, especially for the conditional-distribution and condition-number plots.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the construction is a self-contained composition of triangular conditional maps; score 2 reflects only the presence of a minor, non-load-bearing self-citation.

full rationale

The central derivation is self-contained. Lemmas 3.3 and 3.4 show that lower- and upper-triangular maps qF and pF with qF#μX,Y = pF#μX,Y = μU,F yield conditional samplers Flike and Fpost by substitution. Equations (4)-(5) define S and R explicitly as compositions of these conditional maps, and Theorem 3.7 verifies S^{-1}=R and the push-forward identities by direct change-of-variables computation; the paper transparently states that i) and ii) hold by construction. No fitted parameter is renamed as a prediction: the numerical experiments are sanity checks or illustrations, not out-of-sample predictions forced by their training losses. The only self-citation appearing near the construction is [42], cited for 'Some experimental results using affine maps...', which is not used in the proof; the other self-citations are related-work mentions. The deterministic-likelihood/singular-measure restriction is a genuine assumption/invertibility limitation, but it is a scope condition, not a reduction of the claimed result to its own inputs. Hence no circular step is present; score 2 merely acknowledges a minor self-citation that is not load-bearing.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central theorem is a mathematical construction with no fitted constants; the listed free parameters belong to the illustrative experiments. The axioms are standard measure-theoretic and transport-map assumptions; the key domain assumption is that the joint distribution admits invertible triangular maps, which excludes degenerate (noiseless) likelihoods.

free parameters (3)
  • a,b in toy nonlinear map = not reported
    Define the toy problem in Section 4.2; chosen by hand, not part of the theoretical contribution.
  • Hermite polynomial total order = 4
    Hyperparameter for MParT triangular flow in Section 4.2.
  • Affine normal parameters for MNIST = mean/covariance estimated from training data
    Standard maximum-likelihood fit in Section 4.3.
assumptions (3)
  • domain assumption The joint distribution μU,F admits invertible lower-triangular and upper-triangular maps from a product reference measure μX⊗μY.
    Invoked throughout Section 3; follows from triangular transport theory only for absolutely continuous distributions with non-degenerate conditionals, not for singular/noiseless likelihoods.
  • standard math The reference measure is a product: μX,Y = μX⊗μY.
    Used in Lemma 3.3 and Theorem 3.7; standard in normalizing flows.
  • standard math Existence of regular conditional distributions (disintegration).
    Used to define the likelihood and posterior conditionals in Section 3.1.

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Cite this review

Pith. "Pith review of An invertible generative model for forward and inverse problems." pith.science (2026). https://pith.science/paper/RIPXYMBY

@misc{pith2026250903910,
  author       = {Pith},
  title        = {Pith review of: An invertible generative model for forward and inverse problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RIPXYMBY}},
  note         = {Machine review of arXiv:2509.03910}
}
read the original abstract

We formulate the inverse problem in a Bayesian framework and aim to train a generative model that allows us to simulate (i.e., sample from the likelihood) and do inference (i.e., sample from the posterior). We review the use of triangular normalizing flows for conditional sampling in this context and show how to combine two such triangular maps (an upper and a lower one) in to one invertible mapping that can be used for simulation and inference. We work out several useful properties of this invertible generative model and propose a possible training loss for training the map directly. We illustrate the workings of this new approach to conditional generative modeling numerically on a few stylized examples.

Figures

Figures reproduced from arXiv: 2509.03910 by the authors.

Figure 1
Figure 1. Schematic depiction of the map S for an inpainting problem. In this example, the simulation entails removing part of the digit. The inverse of the map produces an inpainted digit. ally, this leads to a map R : R n`m Ñ R m that will map samples from a reference distribution to samples from the posterior, i.e. U “ RpX,fq with X distributed according to a reference distribution over R n (i.e., standard normal) will res… view at source ↗
Figure 2
Figure 2. Schematic depiction of the maps F, Flike, and Fpost for an inpainting problem (cf. figure 1). A generic map F will map between px, yq and pu,fq (left panel) while the triangular structure allows us to perform conditional sampling pu, yq ÞÑ f or px,fq ÞÑ u by either forward or back substitution (right panel). Lemma 3.4. Let Fp : R n`m Ñ R n`m be an invertible upper-triangular map satis￾fying Fp#µX,Y “ µU,F, for a giv… view at source ↗
Figure 3
Figure 3. Condition numbers of F and S as a function of the noise level σ. }ΣF} Ñ 0 (i.e., a deterministic forward model). Note also that the conditioning of Fp is the same, as both are square roots of ΣU,F. The map S on the other hand, does not degenerate as }ΣF} Ñ 0. If m “ n and K is invertible, for example, we have S “ ˆ 0nˆn ´Inˆn K 0nˆn ˙ , and κpSq “ }K}2. An example is shown in figure 3 for K “ ˆ 2 1 1 2˙ , ΣF “ σ 2 I… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Visualization of the various distributions involved. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Visualization of µU,F and its approximation by F#µX,Y 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Conditional distributions for u “ 0.5 and f “ 1 respectively. The true distribution is depicted in orange, while a histogram of the samples is shown in blue [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Visualisation of the flows induced by F and S [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Condition numbers of ∇Fq, ∇Fp, and ∇S (on a log scale) 16 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 12
Figure 12. Figure 12: figure 12 [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 9
Figure 9. Figure 9: Example of the training data, consisting of handwritten digits from the [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Example of the output of S in simulation mode. The top row shows the input images u while the middle row shows the corresponding true measurements f. The bottom row shows the output of S, showing a good match to the expected output. 17 [PITH_FULL_IMAGE:figures/full_f…
Figure 11
Figure 11. Figure 11: Example of the output of S in inference mode. The top row shows the input images u while the middle row shows the corresponding true measurements f. The bottom row shows the output of S ´1 , showing a good match to the expected output. (a) (b) [PITH_FULL_IMAGE:figure…
Figure 12
Figure 12. Figure 12: Example of the output of S in inference mode. Left we see true digit (’0’) and the corresponding measurement. The posterior mean and pixel-wise standard deviation are shown next to it, indiciting the expected uncertainty regarding the inpainted result. On the right, w…

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.