Optimal Linear Baseline Models for Scientific Machine Learning

Alexander DeLise; Andrea Arnold; Krish Patel; Kyle Loh; Matthias Chung; Meredith Teague

REVIEW 4 major objections 6 minor 117 references

Optimal Linear Baseline Models for Scientific Machine Learning

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read For data generated by a linear forward model with independent zero-mean noise, the best low-rank linear map—the optimal single-layer linear encoder-decoder under mean squared error—is given in closed form by a truncated-SVD formula, for bot

desk verdict Unified closed-form linear baselines; theory is sound, but the SWE validation leaks test information and misapplies the linear-model assumption. read the letter →

arxiv 2508.05831 v1 pith:SRSJXCY3 submitted 2025-08-07 cs.LG cs.NAmath.NA

Alexander DeLise , Kyle Loh , Krish Patel , Meredith Teague , Andrea Arnold , Matthias Chung This is my paper

classification cs.LG cs.NAmath.NA MSC 15A1862C10

keywords scientific machine learninglinear modelslow-rank approximationBayes risk minimizationencoder-decoderautoencoderforward modelinginverse problems

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 tries to establish that the best low-rank linear map from inputs to outputs—the optimal single-layer linear encoder-decoder under mean squared error—is not something you have to train: it has a closed form. For the forward problem of approximating $Y = FX + E$, the minimizer of $E\|AX - Y\|^2$ over $\mathrm{rank}(A) \le r$ is $(F L_X)_r L_X^\dagger$, where $\Gamma_X = L_X L_X^\top$ is a factorization of the input second moment and $(\cdot)_r$ is the truncated SVD. For the inverse problem of recovering $X$ from $Y$, the minimizer is $(\Gamma_X F^\top L_Y^{\dagger\top})_r L_Y^\dagger$. The derivations unify known results—PCA, reduced-rank regression, the Wiener filter, and the classical full-rank affine inverse estimator—by casting them all as special cases of one generalized rank-constrained matrix approximation theorem. If the formulas hold up, every scientific machine learning practitioner gets a parameter-free, provably optimal baseline against which any learned neural network can be judged.

What carries the argument

The load-bearing tool is the generalized rank-constrained matrix approximation theorem of [55]: for $\min_{\mathrm{rank}(W)\le r} \|A - BWC\|_F$, a minimizer of minimal Frobenius norm is $W = B^\dagger (P_B^L A P_C^R)_r C^\dagger$, a pseudoinverse-and-truncated-SVD sandwich. The paper's contribution is the reduction: completing the square in the Bayes risk turns $E\|AX - Y\|^2$ into $\|AL_X - FL_X\|_F^2$ plus a term independent of $A$, and turns $E\|AY - X\|^2$ into $\|AL_Y - C\|_F^2$ with $C = \Gamma_X F^\top L_Y^{\dagger\top}$, putting both problems in exactly the shape the theorem solves. The rank truncation $(\cdot)_r$ imposes the bottleneck, the pseudoinverses $L_X^\dagger$ and $L_Y^\da

What would settle it

For a fully specified linear model (known $\Gamma_X$, $F$, $\Gamma_E$), evaluate the expected loss $E\|AX - (FX+E)\|^2$ analytically and compare the paper's $\widehat A = (F L_X)_r L_X^\dagger$ with a brute-force numerical minimizer over rank-$r$ matrices (projected gradient descent on the true-moment loss, many restarts); any loss gap beyond numerical tolerance falsifies Theorem 2. A second, non-theorem test: apply the inverse formula to data from a known nonlinear map such as $Y = [X_1^2,\ X_1X_2] + E$, or to noise with $E[E \mid X] \neq 0$; Monte Carlo estimation of the loss will show the c

Watch

Extended reading notes

Core claim

Under the linear generative model $Y = FX + E$ with noise $E$ independent of $X$ and zero mean, the paper proves that the forward end-to-end problem $\min_{\mathrm{rank}(A)\le r} E\|AX - (FX+E)\|^2$ is solved by $\widehat A = (F L_X)_r L_X^\dagger$ (Theorem 2), where $\Gamma_X = E[XX^\top] = L_X L_X^\top$ and $(\cdot)_r$ keeps the $r$ largest singular triples. The inverse problem $\min_{\mathrm{rank}(A)\le r} E\|A(FX+E) - X\|^2$ is solved by $\widehat A = (\Gamma_X F^\top L_Y^{\dagger\top})_r L_Y^\dagger$ (Theorem 4), with $\Gamma_Y = F\Gamma_X F^\top + \Gamma_E = L_Y L_Y^\top$. The proofs rewrite the expected squared error as a constant plus a weighted Frobenius problem by completing the sq

Load-bearing premise

Everything rests on the premise that observations actually follow a linear model $Y = FX + E$ with noise $E$ independent of $X$ and zero mean; outside that model the closed forms are heuristics, not proven optima.

Editorial extensions

If this is right

Any single-layer linear encoder-decoder trained on squared loss can match the closed form at best; the formulas therefore give a parameter-free ceiling for linear architectures and a fair benchmark for nonlinear ones.
When the data or the forward operator is rank-deficient, the optimal map simplifies to a projection of $F$ onto the data's column space (e.g., $F U_{L_X,k} U_{L_X,k}^\top$), making explicit which input directions are unrecoverable.
Centering the data switches the formulas from second moments to covariances and adds a bias term ($(F - \widehat A)\mu_X$ forward, $(I_n - \widehat A F)\mu_X$ inverse), so practitioners can choose the form that matches how their data are preprocessed.
The special cases reproduce classical results—PCA for autoencoding, the Wiener filter for denoising, the full-rank affine inverse estimator—so the framework can serve as a single reference point for all of them.
In the shallow-water experiment, the closed-form linear inverse map beats a trained deep nonlinear network, particularly out-of-distribution; if this holds generally, nonlinear architectures need to earn their extra complexity against the linear baseline.

Reading between the lines

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

The gap between the empirical cross-moment $\frac{1}{J}XY^\top$ and $\Gamma_X F^\top$ computed from a fitted linear $F$ is a measurable linearity diagnostic: where it is large, the closed form is not the Bayes optimum, which gives nonlinear models a principled opening and a quantitative reason to prefer them.
The same formulas suggest a training-free way to choose the bottleneck rank: evaluate the closed-form map's held-out error across $r$ at negligible cost and pick the knee, rather than the shallow-water experiment's practice of selecting $r = 250$ by minimizing RMSE over the full dataset, which peeks at test information.
Because the inverse map is the optimal linear reconstruction at every rank, it could serve as a cheap initialization or preconditioner for iterative nonlinear inverse solvers and for Bayesian sampling—a use the paper leaves implicit.
In the financial setting, the optimal affine autoencoder's factors are by construction the best linear summary of asset returns for reconstruction; that makes them a reference point for judging whether nonlinear factor models extract economically different structure or merely overfit noise.

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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

Core math is solid and worth knowing. The paper derives closed-form Bayes-optimal rank-constrained linear maps for forward and inverse problems, plus autoencoding and denoising as special cases, by applying Friedland–Torokhti. The proofs check out; the rank-deficient cases are handled cleanly. This is not a new theorem in the sense that reduced-rank regression and Wiener filtering are known, but the unified treatment under one Bayes-risk umbrella is a useful reference for anyone benchmarking SciML baselines. Section 4.3 is honest that the empirical least-squares solutions appear in prior work, and the code and data are promised on GitHub.

The soft spots are in the experiments, not the algebra. The SWE experiment in Section 5.3 is the load-bearing problem. Theorem 4's optimality assumes Y = FX + E with linear F and independent zero-mean noise. The SWE forward map is nonlinear—the continuity equation retains nonlinear advection terms—so E[XY^T] = Γ_X F^T does not hold with a fixed linear F. The paper replaces it with the empirical cross-covariance; that gives the ordinary empirical linear inverse estimator, not the Bayes-optimal map. The abstract's claim that the theory is validated on nonlinear fluid dynamics overstates what the experiment can support. Additionally, the rank r = 250 is chosen by minimizing RMSE over the entire dataset ('minimal RMSE across the entire dataset'), which leaks test information into model selection. The reported out-of-distribution errors are therefore optimistically biased, and the claim that the optimal linear map performs significantly better than the nonlinear network is not supported by this setup.

The MedMNIST experiments are cleaner and do validate the closed forms against learned networks. The financial section is fine as a demonstration, but the MSE differences between the optimal affine linear map and PCA are small (2.88 vs. 2.96 × 10^-4); calling that 'significantly better' is too strong. The synthetic finance experiment is more direct and gives honest correlations.

There is also a minor gap in Theorem 4's proof: completing the square assumes range(F Γ_X) ⊆ range(Γ_Y), which holds but is never flagged. Easy fix.

Who should read this: people building linear baselines for inverse problems or autoencoders, and anyone comparing learned networks to principled linear estimators. It deserves a serious referee. The theory is sound and useful; the revision needs to fix the SWE experiment—honest rank selection and no claim of Bayes optimality for nonlinear F—and tone down the empirical language. I would not desk-reject it.

Referee Report

4 major / 6 minor

Summary. The paper derives closed-form solutions to rank-constrained, squared-loss estimation problems for linear encoder-decoder architectures in both forward and inverse settings. Theorem 2 gives A = (F L_X)_r L_X^\dagger for min_{rank(A)\le r} E\|AX-(FX+E)\|^2, and Theorem 4 gives A = (\Gamma_X F^\top L_Y^{\dagger\top})_r L_Y^\dagger for the inverse problem min_{rank(A)\le r} E\|AY-X\|^2. Affine variants and special cases (autoencoding, denoising) are provided, recovering PCA and Wiener-filter forms. Numerical experiments cover MedMNIST imaging, financial factor analysis, and a shallow-water-equations inverse problem. The theoretical reduction to Friedland-Torokhti is mostly sound, but the nonlinear PDE experiment is used to claim validation of a theorem whose assumptions it violates, and its rank selection uses the full dataset, including test data.

Significance. If the theorems are correct, the paper offers a useful unification of known linear low-rank estimators and supplies convenient closed-form baselines, with careful handling of rank-deficient data, forward operators, and noise. The MedMNIST experiments confirm that the closed-form mappings outperform gradient-trained linear networks in reaching the empirical loss, and the financial study illustrates a practical baseline application. However, the shallow-water experiment does not validate the Bayes-optimality claim because the linear forward-model assumption is not satisfied, and the reported comparison is further compromised by test-data leakage in rank selection. The theoretical contribution itself is credible and clearly presented, but the empirical overreach in Section 5.3 needs correction before the paper can be accepted.

major comments (4)

[Section 5.3 and Abstract] The SWE experiment applies Theorem 4 to data generated by a nonlinear forward map. Theorem 4 and Eq. (6) require Y = FX + E with a fixed linear F and E independent of X. The simplified SWE model in Appendix C retains nonlinearity through the continuity equation, Eq. (27), which contains products such as (\eta+H)u. Therefore E[XY^\top] = \Gamma_X F^\top does not hold for any fixed linear F. Substituting (1/J)XY^\top for \Gamma_X F^\top, as done in Section 5.3, yields an ordinary empirical linear inverse estimator, not the Bayes-optimal map of Theorem 4. Hence the Abstract's statement that the theory is 'validated' on nonlinear fluid dynamics is unsupported. Please reframe the experiment as an illustrative linear-baseline study, or provide a rigorous argument for why the nonlinearity does not affect the optimality claim.
[Section 5.3, rank selection] The text states: 'Experiments with different values of r demonstrated that the minimal RMSE across the entire dataset was obtained at r \approx 250', and this r is then fixed for all reported experiments. If 'the entire dataset' includes the in-distribution and out-of-distribution test sets described in the same section, then model selection is performed on test data. This leaks test information and biases the reported errors in favor of the linear method. Please choose r on a held-out validation set and report test errors only for a test set not used during model selection.
[Theorem 4, Eq. (15)] The completing-square step in Eq. (15) represents the cross term tr(A F \Gamma_X) as tr(A L_Y C^\top), with C = \Gamma_X F^\top L_Y^{\dagger\top}. This substitution requires L_Y L_Y^\dagger F\Gamma_X = F\Gamma_X, i.e., range(F\Gamma_X) \subseteq range(\Gamma_Y). The condition is true when L_X is a symmetric square root of \Gamma_X and \Gamma_Y = F\Gamma_X F^\top + \Gamma_E, but it is not stated or proved in the manuscript. As written, the proof has a hidden assumption. Please add a short lemma or sentence justifying this range inclusion.
[Section 5.2, Tables 3 and 4] The 'Optimal Affine Linear' autoencoder from Remark 1 is the rank-r PCA reconstruction of centered data, up to an orthogonal rotation of the latent space. Under the same normalization and variance convention, the PCA baseline should give the same reconstruction MSE and the same CEV. Table 3 reports 2.88 vs 2.96 (\times 10^{-4}) and Table 4 reports 0.342 vs 0.333, so either the PCA implementation differs (e.g., uncentered PCA, correlation matrix, or different variance estimator) or there is an inconsistency. Please specify exactly what the PCA baseline computes. If it is standard PCA, reconcile the discrepancy; if it is a different estimator, say so explicitly, because the claimed advantage over PCA is otherwise unclear.

minor comments (6)

[Section 5.3] The sentence 'Recall from Theorem 2 that the optimal linear inverse end-to-end mapping is given by...' should refer to Theorem 4, not Theorem 2.
[Section 4.3, Eq. (21)] The notation '\mu_X \in \mathbb{R}^{784}' is dataset-specific; it should be \mu_X \in \mathbb{R}^n before specializing to the MedMNIST experiment.
[Appendix C, heading] 'experients' should be 'experiments'.
[Table 5] The column headings 'Optimal' and 'Learned' are ambiguous. Use 'Optimal Linear' and 'Learned Nonlinear'.
[Throughout] There are several typographical issues, including 'in of' in the autoencoding paragraph of Section 4.1, 'NRSME' in Section 5.3, 'line data denoising' in Appendix A, and 'fownload' in reference [88]. A careful proofread is recommended.
[Figure 4 caption] The sentence 'with low.' appears incomplete; it should continue with a description of the error magnitude.

Circularity Check

0 steps flagged · score 2.0 of 10

Central closed-form derivations are not circular; SWE validation has validity issues (model misspecification and full-data rank selection) but no equation-level circularity.

full rationale

The paper's main results (Theorems 2 and 4) are derived from a well-defined Bayes-risk minimization by completing the square and applying the external Friedland-Torokhti rank-constrained approximation theorem [55] (quoted as Theorem 1). The optimal mappings are not assumed as inputs; they are consequences of the objective. The special cases (autoencoding, denoising) follow by substituting F=I and recover classical PCA/Wiener results, which the paper explicitly frames as links to prior literature rather than new predictions. Self-citations [36,37,73] are used as motivation and context, not as load-bearing steps in the proofs. The main caveats are empirical, not circular: in Section 5.3 the paper states 'Although we know a priori that the SWEs are nonlinear...' and then approximates E[XY^T] by (1/J)XY^T, which replaces the theorem's identity E[XY^T]=Γ_X F^T with a sample quantity; this is an assumption violation, not a circular derivation. The passage 'Experiments with different values of r demonstrated that the minimal RMSE across the entire dataset was obtained at r ≈ 250' shows rank selection on the full dataset, which biases the reported test comparisons, but again is a data-leakage/validity issue. Because the closed-form theory is self-contained and grounded in an external theorem, no circular step can be exhibited; score 2 reflects minor self-citation presence and these validation caveats.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The ledger shows the central claim rests on the linear observation model, quadratic loss, and Friedland-Torokhti's external theorem. The free parameters are mostly experimental setup choices; the SWE rank selection is the most concerning because it is tuned on the full dataset.

free parameters (5)

bottleneck rank r = r=250 in SWE (selected by minimizing RMSE over full dataset); r=3 in finance; swept in MedMNIST
The rank constraint is the central user-chosen hyperparameter. In the SWE experiment the value is chosen post hoc on the test set, which can inflate reported performance.
ridge regularization term = 1e-2 in SWE; unspecified 'small' in MedMNIST
Added to empirical second-moment matrices to ensure SPD for Cholesky; affects the computed optimal map though usually mildly.
noise standard deviation s_E = 0.05 in MedMNIST
Sets the noise level in the synthetic observation model; the paper does not vary it.
Gaussian blur kernel width s_F = 1.5 in MedMNIST
Defines the forward operator F in the imaging experiment; chosen by the authors.
financial factor count = 3
Latent dimension for the autoencoder, chosen to match the Fama-French three-factor model; not justified by data.

assumptions (6)

domain assumption Observations follow Y = FX + E with E independent of X and zero-mean, with finite second moments (Eq. 1)
This is the data-generating model throughout Section 4; if it fails, the closed-form Bayes-optimal interpretation breaks.
domain assumption Loss is expected squared l2 error (Bayes risk under quadratic loss)
All theorems minimize E||AX - Y||^2; other losses would give different optimal maps.
standard math Theorem 1 (Friedland-Torokhti) is used as a black box
The paper states Theorem 1 and says 'Proof. See [55].' The entire derivation relies on this external theorem.
standard math Second-moment matrices admit symmetric PSD factorizations Gamma_X = L_X L_X^T with L_X full row rank up to rank(Gamma_X)
Every PSD matrix has such a factorization; used throughout Section 4.
domain assumption In the empirical setting, sample moments consistently estimate population moments
The paper replaces Gamma_X, Gamma_Y, Gamma_X F^T with sample statistics; the claimed optimality is asymptotic or approximate.
domain assumption The inverse problem's optimality relies on the linear forward model: E[XY^T] = Gamma_X F^T
Used in Theorem 4 and in the SWE experiment; if the forward process is nonlinear, the derived map is not necessarily Bayes optimal.

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Cite this review

Pith. "Pith review of Optimal Linear Baseline Models for Scientific Machine Learning." pith.science (2026). https://pith.science/paper/SRSJXCY3

@misc{pith2026250805831,
  author       = {Pith},
  title        = {Pith review of: Optimal Linear Baseline Models for Scientific Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRSJXCY3}},
  note         = {Machine review of arXiv:2508.05831}
}

read the original abstract

Across scientific domains, a fundamental challenge is to characterize and compute the mappings from underlying physical processes to observed signals and measurements. While nonlinear neural networks have achieved considerable success, they remain theoretically opaque, which hinders adoption in contexts where interpretability is paramount. In contrast, linear neural networks serve as a simple yet effective foundation for gaining insight into these complex relationships. In this work, we develop a unified theoretical framework for analyzing linear encoder-decoder architectures through the lens of Bayes risk minimization for solving data-driven scientific machine learning problems. We derive closed-form, rank-constrained linear and affine linear optimal mappings for forward modeling and inverse recovery tasks. Our results generalize existing formulations by accommodating rank-deficiencies in data, forward operators, and measurement processes. We validate our theoretical results by conducting numerical experiments on datasets from simple biomedical imaging, financial factor analysis, and simulations involving nonlinear fluid dynamics via the shallow water equations. This work provides a robust baseline for understanding and benchmarking learned neural network models for scientific machine learning problems.

Figures

Figures reproduced from arXiv: 2508.05831 by the authors.

**Figure 1.** End-to-end encoder-decoder architecture. The input x is mapped by Ex onto a latent variable z, then decoded by y ≈ Dz + b to obtain a target vector y. The remainder of this paper will focus on such linear (and correspondingly affine linear) encoder-decoders and autoencoders, and their application to scientific ML problem formulations. We note that in the literature, the terms encoder-decoder and autoencoder are oft… view at source ↗

**Figure 2.** Example of the forward end-to-end process on sample 7181 from ChestMNIST. The left image shows the original input x; the center image shows the blurred version Fx after application of the forward operator F; and the right image shows the final observed measurement y = Fx + ε after addition of Gaussian noise ε. 5.1 MedMNIST Study We begin with a basic biomedical imaging example. For this investigation, we focus prima… view at source ↗

**Figure 3.** Average per-sample ℓ2 reconstruction error versus bottleneck rank r across MedMNIST datasets for both forward and inverse end-to-end problems. Left: Forward process, where the goal is to learn a mapping from x to y. The optimal and learned mappings minimize the empirical losses 1 J PJ j=1 ∥Ab F r xj − yj∥ 2 2 and 1 J PJ j=1 ∥AF r xj − yj∥ 2 2, respectively. Right: Inverse process, where the goal is to reconstruct x … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Example reconstruction results from ChestMNIST with bottleneck rank r = 25 for both forward and inverse end-to-end problems. In all cases, the observed measurement is denoted y and the ground truth is x. Top row: Reconstructions using optimal mappings. Left to right: o…

**Figure 5.** Figure 5: Visualization of the four different kinds of initial conditions used in the dataset. Note that the z axis denotes η (m) for the first and fourth columns, u (m/s) for the second and fifth columns and v (m/s) for the others columns. sion of our stock universe. More gener…

**Figure 6.** Figure 6: Example of actual initial conditions, linear model reconstructions, and nonlinear learned model reconstructions for all three variables, for an instance from the out-ofdistribution testing set. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]

**Figure 7.** Figure 7: Average per-sample ℓ2 reconstruction error versus bottleneck rank r across MedMNIST datasets for the classic noiseless autoencoder problem, where the goal is to learn a rank constrained identity mapping from x to x. The optimal and learned mappings minimize the empiric…

**Figure 8.** Figure 8: Example reconstruction results from ChestMNIST with bottleneck rank r = 25 for the classic linear noiseless autoencoder problem. Top row: Reconstructions using optimal mappings. Left to right: original signal x; optimal reconstruction Ab 25x and corresponding error map…

**Figure 9.** Figure 9: Average per-sample ℓ2 reconstruction error versus bottleneck rank r across MedMNIST datasets in the line data denoising setting. The expressions 1 J PJ j=1 ∥Ab r(xj + εj ) − xj∥ 2 2 and 1 J PJ j=1 ∥Ar(xj + εj ) − xj∥ 2 2 correspond to the optimal and learned reconstruc…

**Figure 10.** Figure 10: Example reconstruction results from ChestMNIST with bottleneck rank r = 25 in the linear data denoising setting. Top row: original ground-truth image x; optimal denoised reconstruction Ab 25(x + ε); and reconstruction error map |Ab 25(x + ε) − x|. Bottom row: noisy ob…

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