A Paired Autoencoder Framework for Inverse Problems via Bayes Risk Minimization

Emma Hart; Julianne Chung; Matthias Chung

REVIEW 3 major objections 5 minor 2 cited by

A Paired Autoencoder Framework for Inverse Problems via Bayes Risk Minimization

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read By training two autoencoders separately on unlabeled data and connecting their latent spaces with a linear map, PAIR builds forward and inverse surrogates for inverse problems that stay accurate when labeled pairs are scarce.

desk verdict A sound Bayes-risk theory for linear paired autoencoders with a clean scarce-pairs story, but the nonlinear claim rests on an unvalidated latent-linearity assumption and thin experiments. read the letter →

arxiv 2501.14636 v1 pith:FBHWXZZ3 submitted 2025-01-24 cs.LG cs.NAmath.NA

Emma Hart , Julianne Chung , Matthias Chung This is my paper · ORCID

classification cs.LGcs.NAmath.NA MSC 65F2265F5568T0768U10

keywords inverseproblemsautoencodersBayesriskminimizationempiricallatent-spacemappinglow-rankapproximationout-of-distributiondetectionsurrogatemodels

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

This paper proposes PAIR, a paired-autoencoder framework for inverse problems. Instead of training one network directly from observations $b$ to parameters $x$, PAIR trains two autoencoders separately on unlabeled data and then fits a linear map between their latent spaces. For fully linear autoencoders the paper proves that the resulting latent maps are Bayes-risk-optimal, meaning they minimize the expected squared loss over the underlying distributions. In the nonlinear, convolutional case PAIR reconstructs blurred MNIST digits more accurately than an end-to-end encoder-decoder network when labeled pairs are scarce. The same architecture yields five cheaply computable metrics that separate in-distribution from out-of-distribution samples, giving a reliability check for new observations.

What carries the argument

The central object is the paired autoencoder: two encoder-decoder networks $(e_x,d_x)$ and $(e_b,d_b)$, trained independently to reconstruct $x$ and $b$ from low-dimensional latent codes, joined by the linear least-squares maps $m = Z_b Z_x^\dagger$ and $m^\dagger = Z_x Z_b^\dagger$ between the latent representations $Z_x$ and $Z_b$ of training samples. In the linear case the autoencoder construction reduces to a rank-constrained problem whose solution is a truncated-SVD projection of the square root of the second-moment matrix $\Gamma = \mathbb{E}[xx^\top]$, so PAIR inherits a PCA/TSVD interpretation as a regularizer. The five PAIR metrics—autoencoding residuals, a forward-surrogate residual, and latent-space mismatches in both directions—are what turn the architecture into an out-of-distribution detector.

What would settle it

Train PAIR on a nonlinear forward operator whose compressed representations are known to be nonlinearly related (e.g., $x \mapsto x^2$ or a parametric wave-equation map) with abundant unlabeled data; if the relative error of the linear bridge $\|Z_b - \hat{M}Z_x\|/\|Z_b\|$ stays large while a nonlinear bridge fits the same codes almost perfectly, the paper's central premise for nonlinear problems is falsified.

Watch

Extended reading notes

Core claim

PAIR builds the forward surrogate $P = d_b \circ m \circ e_x$ and the inverse surrogate $P^\dagger = d_x \circ m^\dagger \circ e_b$, where $(e_x,d_x)$ and $(e_b,d_b)$ are autoencoders learned without pairs and $m,m^\dagger$ are linear maps between latent spaces. In the fully linear setting the paper proves that the optimal latent maps are $M = E_b A \Gamma_x E_x^\top (E_x \Gamma_x E_x^\top)^{-1}$ and $M^\dagger = E_x \Gamma_x^\top A^\top E_b^\top (E_b \Gamma_b E_b^\top)^{-1}$, and that composing them with SVD-based autoencoders gives the compressed surrogates $\tilde{P} = U_{L_b,r_b}U_{L_b,r_b}^\top A U_{L_x,r_x}U_{L_x,r_x}^\top$ and $\tilde{P}^\dagger = U_{L_x,r_x}\Sigma_{L_x,r_x}^{2}U_{L_x,r_x}^\top A^\top U_{L_b,r_b}\Sigma_{L_b,r_b}^{-2}U_{L_b,r_b}^\top$. The component maps are each Bayes-risk-optimal, although the composed surrogates are not themselves claimed to minimize the reconstruction risk. With samples, the same maps become the Moore-Penrose solutions $\hat{M} = Z_b Z_x^\dagger$ and $\hat{M}^\dagger = Z_x Z_b^\dagger$. The paper demonstrates on MNIST deblurring that this architecture outperforms an end-to-end encoder-decoder when supervised pairs are few, while the end-to-end network keeps a slight edge when all 60,000 pairs are used.

Load-bearing premise

The load-bearing premise is that a straight-line (linear) relationship between the compressed internal representations learned by the two autoencoders is enough to capture the forward and inverse relationships, a claim the paper verifies empirically only for convolutional autoencoders on blurred digits; if that linearity fails for a more complex forward operator, the scarce-pairs advantage for nonlinear inverse problems is unsupported.

Editorial extensions

If this is right

With abundant unlabeled observations and parameters but few paired samples, PAIR produces smaller average reconstruction error on the MNIST deblurring test set than a comparably sized end-to-end encoder-decoder network.
In the fully linear, noiseless case with full latent ranks, PAIR's forward surrogate equals $A$ and its inverse surrogate equals $A^{-1}$; with truncation, it acts as $A$ on the data subspace spanned by the training samples.
The linear PAIR theory contains PCA and truncated SVD as special cases, giving PAIR an inherent regularizing effect that removes the need to pick a variational regularization parameter.
For a new observation, the five PAIR metrics computed without access to the true solution separate in-distribution MNIST samples from out-of-distribution notMNIST samples, so they can flag predictions that are likely unreliable.
Because the autoencoders are trained independently and in parallel, PAIR can use different-sized unlabeled datasets for the observation and parameter spaces, and the parameter autoencoder can be reused when the forward operator changes.

Reading between the lines

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

If the empirical linearity of latent-space maps holds beyond imaging, PAIR doubles as an adjoint-free surrogate: the same linear maps provide forward and adjoint actions, which the paper names only as future work.
The residual of the linear bridge, $\|Z_b - \hat{M}Z_x\|/\|Z_b\|$, could serve as a continuous confidence score; thresholds taken from training data would turn the five qualitative OOD metrics into a calibrated rejection rule.
Because the latent ranks $r_x$ and $r_b$ are free hyperparameters, PAIR's regularization strength is effectively selected by rank; the paper validates ranks by test-error curves, but the OOD metrics could automate rank selection.
The semiconvergence seen in the PAIR inverse surrogate at rank about 2,600 in the CT experiment suggests PAIR regularizes like truncated SVD with data-informed subspaces; a testable consequence is that the optimal ranks will track the observation noise level, as they do for TSVD.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my take. The genuinely new thing in this paper is the theoretical core: Theorems 3.2 and 3.3 give closed-form Bayes-risk-optimal linear maps between autoencoder latent spaces, and Proposition 3.4 gives the empirical analog. I checked the algebra; it's clean, assumptions are stated, and the paper is honest that composing individually optimal components does not give a jointly optimal surrogate. The reduction of the fully linear PAIR to PCA plus least-squares is a helpful unifying perspective, and the connection to TSVD with data-dependent subspaces is worth having. In the linear CT example, the data-driven subspaces do beat TSVD, which is a credible demonstration.

Where I get concerned is the nonlinear case. The entire scarce-pairs advantage for MNIST deblurring depends on the claim that a linear map between independently trained autoencoder latent spaces suffices. The paper points to [18, 20] for that, but never checks it for its own CNN autoencoders. A simple diagnostic—compare the linear latent map against a small MLP, or measure how well the latent coordinates are linearly related—would settle it. Without that, the nonlinear results are an anecdote, not a theorem. The experiment itself is also thin: one forward model, one small architecture, one train/test split, no repeated runs, and no comparison against the closest prior methods ([15], [19]) despite the abstract claiming to outperform 'existing approaches.' The reported 0.3783 error is not compared against the autoencoder reconstruction floor, so we can't tell how much of the error is the latent map versus the autoencoders.

The OOD metrics are a nice byproduct—cheap, intuitive, and they separate MNIST from notMNIST well. But that's a single easy case.

The paper doesn't ship code at submission ('will be provided' is not enough for a numerical paper). That's a concrete fix.

All that said, the theoretical contribution is real and the writing is clear. The paper deserves a serious referee. I'd recommend conditional acceptance after (a) shipping code, (b) adding a latent-linearity check, (c) adding [15] and [19] as baselines, and (d) toning down the abstract's 'outperforms existing approaches' to match the actual evidence.

Referee Report

3 major / 5 minor

Summary. The paper proposes PAIR (Paired Autoencoders for Inference and Regularization), a framework that trains two autoencoders separately on the parameter space x and the observation space b and then fits linear maps between their latent codes to form forward and inverse surrogates P = db∘m∘ex and P† = dx∘m†∘eb. The theoretical part derives Bayes-risk and empirical-Bayes-risk optimality results for fully linear autoencoders: Theorem 3.1 identifies the optimal linear autoencoder with a truncated-SVD projection, Theorems 3.2 and 3.3 give closed-form optimal latent maps and the resulting compressed surrogates, and Proposition 3.4 gives the empirical-data analog. The numerical part applies the linear PAIR to CT reconstruction (comparing with TSVD) and a nonlinear CNN PAIR to MNIST deblurring (comparing with an end-to-end encoder-decoder under varying numbers of paired training samples). The paper also proposes five cheaply computable metrics for out-of-distribution detection and demonstrates them on notMNIST.

Significance. The linear theory is a genuine contribution: the derivations in Theorems 3.2, 3.3, and Proposition 3.4 are clean, the assumptions (SPD second moments, full row rank) are stated, and the resulting formulas connect the framework to classical low-rank approximation and pseudo-inverse computations. The paper is also honest that the composed surrogates are not jointly optimal even when each component is. However, the broader practical claim — that PAIR outperforms supervised end-to-end approaches when paired training data are scarce — is only demonstrated for nonlinear autoencoders in a narrow setting and rests on the unvalidated premise that linear maps between independently learned latent spaces suffice for nonlinear encoders. The out-of-distribution metrics are a useful secondary contribution but are evaluated only qualitatively. If the nonlinear latent-linearity assumption can be validated and the empirical comparison broadened, the paper would be a solid contribution to data-driven inverse problems.

major comments (3)

[Section 4.2, Eq. (3.17)] The central scarce-pairs advantage for nonlinear problems rests on the assumption that a linear map between the latent spaces of the independently trained CNN autoencoders is adequate: cM = Zb Zx† is optimal only as linear regression on fixed latent coordinates. Theorems 3.2 and 3.3 cover only linear autoencoders, and Section 1's appeal to Refs. [18,20] is not a substitute for validating this assumption on the paper's own architecture. Please add a quantitative check of latent linearity (for example, the relative residual of the linear fit cMZx−Zb versus the total latent variance, or a nonlinear latent-map baseline) and at least one additional forward operator; without that, the nonlinear PAIR advantage should be reported as a heuristic rather than as a general result.
[Figure 6 / Section 4.2.1] The empirical comparison uses one forward model (Gaussian blur, 8×8 kernel with σ=10), one 236-parameter autoencoder architecture, and no error bars or multiple random seeds. The abstract's statement that PAIR 'outperforms existing approaches' is overbroad because the prior paired-autoencoder methods cited in the introduction (Refs. [15,18,19]) are not included as baselines in Figure 6. Please add those baselines or restrict the claim to the specific comparison made, and report variability across seeds or data splits.
[Section 4.2.1] The reported average relative reconstruction error of 0.3783 for the MNIST PAIR network is not compared against the autoencoder reconstruction floor, ||dx(ex(x))−x||2/||x||2. Without this decomposition it is unclear whether the error is dominated by the autoencoders or by the linear latent map. If the floor is already near 0.38, the latent map is not the limiting factor and the comparison in Figure 6 is hard to interpret; if the floor is much smaller, the gap identifies the latent-map error. Please report both components.

minor comments (5)

[Eq. (3.16)] The term U⊤_Lrx in the expression for eP† appears to be a typo; it should presumably be U⊤_Lx,rx to match the other factors.
[Section 3.2.1] The decoder dimension for the b autoencoder is written as Db ∈ Rq×rq with rb ≤ q; the subscript rq should be rb.
[Figure 8] The first metric is defined in Section 4.2.2 as ||(db∘eb)(b)−b||2/||b||2, but the Figure 8 axis label omits the subtraction of b; please make the label consistent with the definition.
[Section 4.2.1] The learning-rate schedule is described as 10^-3, 10^-4, 10^-3, 10^-4 over the four hundred epochs; if this is intentional (rather than a typo for a monotonically decreasing schedule), please clarify the rationale.
[Section 3.1.1] The term 'Kulback-Leibner' should be 'Kullback-Leibler'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: PAIR's linear optimality results follow from direct least-squares/SVD algebra, and numerical claims are evaluated on held-out test data; the main author-overlap citation is an external low-rank theorem, not a self-referential input.

full rationale

I walked the paper's derivation chain. Section 3.1 reduces the linear autoencoder problem to min_{rank(Y)<=r} ||YL - L||_F (Eq. 3.5), and Theorem 3.1 gives the SVD solution by citing the previously published rank-constrained result [11]. Section 3.2.1 derives the latent maps M and M-dagger in Theorem 3.2 by rewriting the least-squares objectives as ||(MEx - EbA)Lx||_F and a similar Frobenius-norm expression; the solutions are direct Moore-Penrose identities, not the target result assumed. Theorem 3.3 composes those maps with the SVD encoder/decoder forms (3.14), and the proof algebra is explicit and self-contained. Proposition 3.4 follows the same pattern for the empirical case. No fitted parameter is relabeled as a prediction: the autoencoders are trained on unlabeled data, the latent map is fitted on paired data, and the reported errors are computed on held-out test sets (Figures 3 and 6). The nonlinear section's reliance on linear latent maps is an empirical premise attributed to external works [18,20], and it is a generalizability/correctness concern rather than a circular reduction of the paper's own claim to its definition. The author-overlap citation [11], used to prove Theorem 3.1, is a published rank-constrained least-squares theorem whose stated assumptions do not include the present autoencoder target; it is therefore independent evidence, not circular self-support. No step in the derivation is equivalent by construction to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central derivations rest on standard second-moment assumptions and full-rank conditions, all stated in the text. The theoretical claims require no invented entities and, in the linear setting, no fitted constants: the only hand-chosen numbers are the latent ranks and experimental setup constants, both swept or fixed transparently. The nonlinear experiments add the unproven assumption that latent-space maps are linear.

free parameters (4)

target latent dimension rx = swept 1-4096 (CT); 147 (MNIST)
User-selected compression rank for the parameter space; error curves are shown over the whole sweep, and the 'best' inverse rank about 2600 is read from the test-error curve, so no constant is hidden.
input latent dimension rb = swept 1-3240 (CT); 147 (MNIST)
User-selected compression rank for the observation space, treated identically to rx.
deblurring task constants = 8x8 Gaussian blur, sigma=10; noise variance 0.01
Hand-chosen experimental setup (Section 4.2) that defines the difficulty of the MNIST task; not determined by theory and not fitted, but load-bearing for the reported error magnitudes.
MNIST autoencoder architecture and training schedule = 5 conv layers (2,3,3,2,1 channels), 236 parameters, 400 epochs, ADAM, batch 256
The paper says the architecture is 'intentionally kept simple to highlight the core concepts'; results may shift with architecture quality, which the paper acknowledges in Section 4.2.1.

assumptions (5)

domain assumption X has SPD second moment Γx = LxLx^T; noise ε is independent of X with zero mean and SPD covariance Γε
Invoked in Section 3.2.1 to derive the Bayes-risk objective and the optimal latent maps in Theorem 3.2. These are standard modeling assumptions but restrict the theory to second-order statistics.
domain assumption Encoders Ex and Eb have full row rank; in the empirical setting σrx(X) > 0 and σrb(B) > 0
Needed to invert ExΓxEx^T and EbΓbEb^T in Theorem 3.2 and to define cM = ZbZx† in Eq. (3.17). Stated as assumptions, not verified in the experiments.
domain assumption Linear maps between latent spaces suffice for nonlinear autoencoders
The CNN experiments in Section 4.2 use the linear bridge cM = ZbZx† between independently trained autoencoders without a supporting theorem; the paper cites the empirical observation of [18, 20] that such maps are linear. This is the main fragile premise for the nonlinear claims.
standard math Eckart-Young-type low-rank approximation result
Theorem 3.1 is stated to follow 'directly from Theorem 3.1 in [11]', a self-citation; the underlying low-rank approximation theorem is classical, so the dependency is benign but worth flagging.
domain assumption Test samples are drawn from the same distribution as training data
Test errors in Section 4 are computed on held-out sets generated like the training data; Section 4.2.2 studies the failure mode explicitly with notMNIST.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Paired Autoencoder Framework for Inverse Problems via Bayes Risk Minimization." pith.science (2026). https://pith.science/paper/FBHWXZZ3

@misc{pith2026250114636,
  author       = {Pith},
  title        = {Pith review of: A Paired Autoencoder Framework for Inverse Problems via Bayes Risk Minimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBHWXZZ3}},
  note         = {Machine review of arXiv:2501.14636}
}

read the original abstract

In this work, we describe a new data-driven approach for inverse problems that exploits technologies from machine learning, in particular autoencoder network structures. We consider a paired autoencoder framework, where two autoencoders are used to efficiently represent the input and target spaces separately and optimal mappings are learned between latent spaces, thus enabling forward and inverse surrogate mappings. We focus on interpretations using Bayes risk and empirical Bayes risk minimization, and we provide various theoretical results and connections to existing works on low-rank matrix approximations. Similar to end-to-end approaches, our paired approach creates a surrogate model for forward propagation and regularized inversion. However, our approach outperforms existing approaches in scenarios where training data for unsupervised learning are readily available but training pairs for supervised learning are scarce. Furthermore, we show that cheaply computable evaluation metrics are available through this framework and can be used to predict whether the solution for a new sample should be predicted well.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Variational Sparse Paired Autoencoders (vsPAIR) for Inverse Problems and Uncertainty Quantification
cs.LG 2026-02 conditional novelty 5.0 of 10

vsPAIR couples a Gaussian VAE over observations with a spike-and-slab sparse VAE over the quantity of interest via a learned latent mapping, yielding fast inverse reconstructions whose active latent dimensions can be ...
Optimal Linear Baseline Models for Scientific Machine Learning
cs.LG 2025-08 conditional novelty 5.0 of 10

Closed-form rank-constrained linear estimators, derived from Bayes risk, unify forward modeling, inverse recovery, autoencoding, and denoising, and often match or beat trained neural networks.

Reference graph

Works this paper leans on

43 extracted references · 36 canonical work pages · cited by 2 Pith papers

[15]

Chung, E

M. Chung, E. Hart, J. Chung, B. Peters, and E. Haber, Paired autoencoders for likelihood- free estimation in inverse problems , Machine Learning: Science and Technology, 5 (2024), p. 045055

work page 2024
[19]

Y. Feng, Y. Chen, P. Jin, S. Feng, Z. Liu, and Y. Lin, Auto-linear phenomenon in subsurface imaging, 2024, https://arxiv.org/abs/2305.13314

work page Pith review arXiv 2024
[1]

B. M. Afkham, J. Chung, and M. Chung , Learning regularization parameters of inverse problems via deep neural networks , Inverse Problems, 37 (2021), p. 105017

work page 2021
[2]

A. C. Antoulas, C. A. Beattie, and S. G¨u˘gercin, Interpolatory methods for model reduction, SIAM, 2020

work page 2020
[3]

Arridge, P

S. Arridge, P. Maass, O. ¨Oktem, and C.-B. Sch ¨onlieb, Solving inverse problems using data-driven models, Acta Numerica, 28 (2019), pp. 1–174

work page 2019
[4]

Atkinson, A

A. Atkinson, A. Donev, and R. Tobias , Optimum experimental designs, with SAS , vol. 34, This manuscript is for review purposes only. PAIRED AUTOENCODERS 21 OUP Oxford, 2007

work page 2007
[5]

Y. Bai, W. Chen, J. Chen, and W. Guo , Deep learning methods for solving linear inverse problems: Research directions and paradigms , Signal Processing, 177 (2020), p. 107729, https://doi.org/10.1016/j.sigpro.2020.107729

arXiv 2020
[6]

Baldi and K

P. Baldi and K. Hornik , Neural networks and principal component analysis: Learning from examples without local minima , Neural Networks, 2 (1989), pp. 53–58

work page 1989

Show all 43 references

[7]

X. Bao, J. Lucas, S. Sachdeva, and R. B. Grosse , Regularized linear autoencoders recover the principal components, eventually, Advances in Neural Information Processing Systems, 33 (2020), pp. 6971–6981

2020
[8]

Benner, S

P. Benner, S. Gugercin, and K. Willcox , A survey of projection-based model reduction methods for parametric dynamical systems , SIAM Review, 57 (2015), pp. 483–531

2015
[9]

Bourlard and Y

H. Bourlard and Y. Kamp , Auto-association by multilayer perceptrons and singular value decomposition, Biological cybernetics, 59 (1988), pp. 291–294

1988
[10]

Bulatov , NotMNIST dataset , Google (Books/OCR), Tech

Y. Bulatov , NotMNIST dataset , Google (Books/OCR), Tech. Rep.[Online]. Available: http://yaroslavvb. blogspot. it/2011/09/notmnist-dataset. html, 2 (2011), p. 4

2011
[11]

Chung and M

J. Chung and M. Chung , Optimal regularized inverse matrices for inverse problems , SIAM Journal on Matrix Analysis and Applications, 38 (2017), pp. 458–477, https://doi.org/10. 1137/16M1066531

2017
[12]

Chung and S

J. Chung and S. Gazzola, Computational methods for large-scale inverse problems: A survey on hybrid projection methods , SIAM Review, 66 (2024), pp. 205–284, https://doi.org/10. 1137/21M1441420

2024
[13]

Chung , Randomized Shepp-Logan phantom , 2020, https://github.com/matthiaschung/ Random-Shepp-Logan-Phantom

M. Chung , Randomized Shepp-Logan phantom , 2020, https://github.com/matthiaschung/ Random-Shepp-Logan-Phantom

2020
[14]

Chung, R

M. Chung, R. Archibald, P. Atzberger, and J. M. Solomon , Sparse L1-autoencoders for scientific data compression , arXiv preprint arXiv:2405.14270, (2024)

2024 arXiv
[16]

Deng , The MNIST database of handwritten digit images for machine learning research , IEEE Signal Processing Magazine, 29 (2012), pp

L. Deng , The MNIST database of handwritten digit images for machine learning research , IEEE Signal Processing Magazine, 29 (2012), pp. 141–142

2012
[17]

I. G. Enting , Inverse problems in atmospheric constituent transport , Cambridge University Press, 2002

2002
[18]

Y. Feng, Y. Chen, S. Feng, P. Jin, Z. Liu, and Y. Lin , An intriguing property of geophysics inversion, in International Conference on Machine Learning, PMLR, 2022, pp. 6434–6446

2022
[20]

Y. Feng, Y. Chen, Y. Lee, and Y. Lin , On a hidden property in computational imaging , 2024, https://arxiv.org/abs/2410.08498

2024 arXiv
[21]

Gazzola, P

S. Gazzola, P. C. Hansen, and J. G. Nagy , IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems , Numerical Algorithms, 81 (2019), pp. 773–811

2019
[22]

H. Goh, S. Sheriffdeen, J. Wittmer, and T. Bui-Thanh, Solving Bayesian inverse problems via variational autoencoders, 2021, https://arxiv.org/abs/1912.04212

2021 arXiv
[23]

F. J. Gonzalez and M. Balajewicz , Deep convolutional recurrent autoencoders for learn- ing low-dimensional feature dynamics of fluid systems , 2018, https://arxiv.org/abs/1808. 01346

2018
[24]

Goodfellow, Y

I. Goodfellow, Y. Bengio, and A. Courville , Deep Learning, MIT Press, Boston, 2016

2016
[25]

Hadamard, Lectures on Cauchy’s Problem in Linear Differential Equations, Yale University Press, New Haven, 1923

J. Hadamard, Lectures on Cauchy’s Problem in Linear Differential Equations, Yale University Press, New Haven, 1923

1923
[26]

P. C. Hansen , Discrete Inverse Problems: Insight and Algorithms , SIAM, 2010

2010
[27]

G. E. Hinton and R. R. Salakhutdinov , Reducing the dimensionality of data with neural networks, science, 313 (2006), pp. 504–507

2006
[28]

Kovachki, Z

N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, and A. Anandkumar , Neural operator: Learning maps between function spaces , 2023, https: //arxiv.org/abs/2108.08481

2023 arXiv
[29]

Kulkarni, S

K. Kulkarni, S. Lohit, P. Turaga, R. Kerviche, and A. Ashok , Reconnet: Non-iterative reconstruction of images from compressively sensed measurements , in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016

2016
[30]

S. Lan, S. Li, and B. Shahbaba , Scaling up Bayesian uncertainty quantification for inverse problems using deep neural networks , SIAM/ASA Journal on Uncertainty Quantification, 10 (2022), pp. 1684–1713, https://doi.org/10.1137/21M1439456

2022 doi
[31]

Lee and K

K. Lee and K. T. Carlberg , Model reduction of dynamical systems on nonlinear mani- This manuscript is for review purposes only. 22 E. HART, J. CHUNG, AND M. CHUNG folds using deep convolutional autoencoders, Journal of Computational Physics, 404 (2020), p. 108973

2020
[32]

H. Li, J. Schwab, S. Antholzer, and M. Haltmeier , NETT: Solving inverse problems with deep neural networks , Inverse Problems, 36 (2020), p. 065005

2020
[33]

Lucas, M

A. Lucas, M. Iliadis, R. Molina, and A. K. Katsaggelos , Using deep neural networks for inverse problems in imaging: Beyond analytical methods , IEEE Signal Processing Maga- zine, 35 (2018), pp. 20–36, https://doi.org/10.1109/MSP.2017.2760358

2018
[34]

Pichi, B

F. Pichi, B. Moya, and J. S. Hesthaven , A graph convolutional autoencoder approach to model order reduction for parametrized PDEs , Journal of Computational Physics, 501 (2024), p. 112762

2024
[35]

Piening and M

M. Piening and M. Chung , Paired Wasserstein autoencoders for conditional sampling , arXiv preprint arXiv:2412.07586, (2024)

2024
[36]

Plaut, From principal subspaces to principal components with linear autoencoders , arXiv preprint arXiv:1804.10253, (2018)

E. Plaut, From principal subspaces to principal components with linear autoencoders , arXiv preprint arXiv:1804.10253, (2018)

2018 arXiv
[37]

Pukelsheim, Optimal design of experiments , SIAM, 2006

F. Pukelsheim, Optimal design of experiments , SIAM, 2006

2006
[38]

Salakhutdinov, A

R. Salakhutdinov, A. Mnih, and G. Hinton , Restricted Boltzmann machines for collabo- rative filtering, in Proceedings of the 24th International Conference on Machine learning, 2007, pp. 791–798

2007
[39]

Sanz-Alonso, A

D. Sanz-Alonso, A. Stuart, and A. Taeb, Inverse problems and data assimilation , vol. 107, Cambridge University Press, 2023

2023
[40]

Torralba, R

A. Torralba, R. Fergus, and Y. Weiss , Small codes and large image databases for recogni- tion, in 2008 IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2008, pp. 1–8

2008
[41]

Ucinski , Optimal measurement methods for distributed parameter system identification , CRC press, 2004

D. Ucinski , Optimal measurement methods for distributed parameter system identification , CRC press, 2004

2004
[42]

W ang, Y

H. W ang, Y. Chen, J. Kang, Y. Wu, Y. J. Kim, and Y. Lin , Wavediffusion: Exploring full waveform inversion via joint diffusion in the latent space , 2024, https://arxiv.org/abs/ 2410.09002

2024
[43]

W ang, H

Y. W ang, H. Yao, and S. Zhao, Auto-encoder based dimensionality reduction, Neurocomput- ing, 184 (2016), pp. 232–242. This manuscript is for review purposes only

2016

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [15]

Chung, E

M. Chung, E. Hart, J. Chung, B. Peters, and E. Haber, Paired autoencoders for likelihood- free estimation in inverse problems , Machine Learning: Science and Technology, 5 (2024), p. 045055

work page 2024

[2] [19]

Y. Feng, Y. Chen, P. Jin, S. Feng, Z. Liu, and Y. Lin, Auto-linear phenomenon in subsurface imaging, 2024, https://arxiv.org/abs/2305.13314

work page Pith review arXiv 2024

[3] [1]

B. M. Afkham, J. Chung, and M. Chung , Learning regularization parameters of inverse problems via deep neural networks , Inverse Problems, 37 (2021), p. 105017

work page 2021

[4] [2]

A. C. Antoulas, C. A. Beattie, and S. G¨u˘gercin, Interpolatory methods for model reduction, SIAM, 2020

work page 2020

[5] [3]

Arridge, P

S. Arridge, P. Maass, O. ¨Oktem, and C.-B. Sch ¨onlieb, Solving inverse problems using data-driven models, Acta Numerica, 28 (2019), pp. 1–174

work page 2019

[6] [4]

Atkinson, A

A. Atkinson, A. Donev, and R. Tobias , Optimum experimental designs, with SAS , vol. 34, This manuscript is for review purposes only. PAIRED AUTOENCODERS 21 OUP Oxford, 2007

work page 2007

[7] [5]

Y. Bai, W. Chen, J. Chen, and W. Guo , Deep learning methods for solving linear inverse problems: Research directions and paradigms , Signal Processing, 177 (2020), p. 107729, https://doi.org/10.1016/j.sigpro.2020.107729

arXiv 2020

[8] [6]

Baldi and K

P. Baldi and K. Hornik , Neural networks and principal component analysis: Learning from examples without local minima , Neural Networks, 2 (1989), pp. 53–58

work page 1989