Proper Latent Decomposition

Daniel Kelshaw; Luca Magri

REVIEW 4 major objections 6 minor 24 references

Proper Latent Decomposition

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Proper latent decomposition generalizes POD to curved manifolds, recovering physical modes from autoencoder latent spaces.

desk verdict A plausible nonlinear extension of POD with nice demos, undercut by overclaims and unquantified metric conditioning; deserves review but needs major revision. read the letter →

arxiv 2412.00785 v1 pith:6ISXZO37 submitted 2024-12-01 cs.LG physics.flu-dyn

Daniel Kelshaw , Luca Magri This is my paper · ORCID

classification cs.LGphysics.flu-dyn MSC 68T0758A0535Q30

keywords properlatentdecompositionorthogonalnonlinearreduced-ordermodelingautoencodersRiemanniangeometryEikonalequationprincipalgeodesicsKolmogorovflow

topics Navier-Stokes Regularity

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 introduces proper latent decomposition (PLD), a method that extends proper orthogonal decomposition (POD) from flat Euclidean spaces to the curved manifolds that autoencoders learn for high-dimensional data. The paper shows that by treating the latent space as a Riemannian manifold and using its inferred metric tensor, one can compute geodesic distance, find a meaningful mean, and extract energy-ordered principal geodesic modes. These modes, unlike linear POD modes, can correspond to physical structures in the underlying flow. The authors demonstrate this on a laminar bluff-body wake, where they obtain a semi-analytical description of the flow, and on turbulent Kolmogorov flow, where the leading PLD mode decodes to a physical flow field. The work matters because it offers a principled way to make autoencoder latent spaces interpretable for nonlinear reduced-order modeling.

What carries the argument

The central object is the Riemannian manifold structure of the autoencoder latent space, determined by the metric tensor inferred from the decoder. A metric-constrained Eikonal solver yields the distance function on the manifold, whose gradient flow gives geodesics; those geodesics support the Fréchet mean and a robust log map via multiple shooting with a continuous adjoint. Once data are mapped to the tangent space at the mean, standard SVD produces energy-ordered modes, and the exponential map returns them to the manifold as principal geodesics.

What would settle it

Train a standard autoencoder on a dataset lying on a manifold with known geodesics, such as a sphere or torus embedded in three dimensions, compute PLD modes without metric regularization, and compare the decoded principal geodesic trajectories to the ground-truth geodesics; substantial deviation or failure of the log-map solver to converge would confirm the dependence on metric conditioning.

Watch

Extended reading notes

Core claim

PLD computes, on the latent manifold inferred by an autoencoder, the Fréchet mean of the data, maps the encoded samples to the tangent space at that mean, performs singular value decomposition to obtain an orthonormal, energy-ordered basis, and maps the principal components back to the manifold via the exponential map to obtain principal geodesics. The paper claims these geodesics maximize variance of the data on the manifold, generalize POD to nonlinear settings, and provide interpretable, physical modes. For the laminar wake, the tangent-space data lie on a simple ellipse, yielding a semi-analytical solution of the Navier-Stokes flow with a single angular variable; for the turbulent Kolmogorov flow, the leading PLD mode reveals coherent vortical structures that differ from linear POD modes and resemble a physical realization of the governing equations.

Load-bearing premise

The whole method rests on the autoencoder's inferred metric tensor being well-conditioned enough to support reliable distance and geodesic computations; if that condition fails, the principal geodesic modes are corrupted.

Editorial extensions

If this is right

For periodic or quasi-periodic flows, PLD can produce low-dimensional semi-analytical reduced-order models, as demonstrated by the single-variable ellipse model of the laminar wake.
PLD provides a nonlinear alternative to POD that can represent data lying on curved manifolds more faithfully than linear modes.
Regularizing the autoencoder's inferred metric improves numerical conditioning, making geodesic computation stable and potentially enabling higher-dimensional latent spaces.
Decoded principal geodesic modes give interpretable, physical coordinates, turning autoencoder latent spaces from opaque coordinates into analyzable dynamical features.

Reading between the lines

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

A natural extension is to use the geodesic coordinates as a dynamical system, evolving the principal-mode amplitudes to forecast non-periodic turbulent flows, which the paper does not explicitly pursue.
Because PLD modes are energy-ordered by construction, they could serve as a curved-manifold analog of a Galerkin basis for reduced-order models, a connection the paper leaves implicit.
A direct synthetic test on a manifold with known geodesics, such as a sphere or torus embedded in three dimensions, would isolate the error of the Eikonal solver and log map from the error of the autoencoder itself.

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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

The paper is a genuine integration of existing pieces rather than a new theoretical breakthrough. What is new is the end-to-end pipeline: infer a latent manifold with an autoencoder, compute geodesic distances with a learned Eikonal solver, then extract principal geodesics by SVD in the tangent space at the Fréchet mean. Two fluid cases show the pipeline works well enough to produce plausible modes, and for the laminar wake the latent structure really does look like an ellipse, which is a nice sanity check. That is real work, and the demonstrations are the best part.

But the paper oversells itself. Calling this the first manifold-aware nonlinear ROM is wrong; principal geodesic analysis has been around for decades and is not cited. The 'semi-analytical solution of Navier-Stokes' is an ellipse fit to the tangent-space data, not a derivation. The claim that the Kolmogorov mode is a 'physical realization of the governing equations' rests on visual resemblance. The central numerical issue—conditioning of the autoencoder-inferred metric—is acknowledged but never quantified for the actual models that produced the results. Figure 6 shows a 'prototypical' autoencoder, not the production one. That matters because every downstream step (mean, log map, SVD, exp map) uses the metric and its inverse. If the metric is badly conditioned, the modes could be numerical artifacts. This is a genuine soft spot, not a manufactured one.

The lack of code and data makes independent checking harder, but that is a secondary concern. The method is plausible and the demos are suggestive. The paper deserves a serious referee, but it needs major revision: temper the claims, cite the PGA literature, quantify conditioning for the actual trained autoencoders, and ideally release code.

For a reading group, it is worth discussing as an example of applying geometric statistics to fluid problems, but not as a settled result. I would send it to review.

Referee Report

4 major / 6 minor

Summary. The paper introduces proper latent decomposition (PLD), a method that generalizes proper orthogonal decomposition (POD) to nonlinear manifolds. The pipeline consists of: (i) training a convolutional autoencoder to map high-dimensional fluid data to a low-dimensional latent manifold and inferring a Riemannian metric from the decoder; (ii) computing geodesic distances and a Fréchet mean using a metric-constrained Eikonal solver; (iii) mapping data to the tangent space at the mean via a log-map that combines gradient-based trajectory interpolation with multiple shooting; and (iv) performing SVD in the tangent space and mapping the principal components back to the manifold via the exponential map. The method is demonstrated on a laminar bluff-body wake and a two-dimensional turbulent Kolmogorov flow. For the laminar case, the tangent-space data are fit with an ellipse, yielding a single-variable description of the flow. For the Kolmogorov flow, the leading geodesic mode is visualized and claimed to be a physical realization of the governing equations.

Significance. If the claims hold, PLD offers a principled, manifold-aware extension of POD that could be valuable for interpretable nonlinear reduced-order modeling. The paper is commendable for being explicit about the numerical challenges (Section 7.3) and for proposing a geometric regularization to stabilize the metric. However, the verification is almost exclusively qualitative: there are no error bars, no quantitative comparison with POD, no conditioning diagnostics for the production models, and no reconstruction-error or energy metrics for the modes. The 'semi-analytical expression' for the laminar wake is a fitted ellipse, not a derived solution. These gaps leave the central claim plausible but not yet established.

major comments (4)

[Section 7.3] The paper's central pipeline (Sections 3–6) relies on the metric tensor g inferred by the autoencoder being well-conditioned on the data support, because the Eikonal solver (Section 5), the log-map shooting (Section 6), and the Fréchet mean (Eq. 3.1) all involve g or g^{-1}. Section 7.3 concedes that standard autoencoder training produces metrics with 'exceedingly large' condition numbers and exponential magnification factors, and states that regularization was introduced to stabilize the metric. Yet no quantitative conditioning or magnification-factor statistics are reported for the actual autoencoders used in Sections 7.1 and 7.2; Figure 6 shows only a 'prototypical' autoencoder trained on the Kolmogorov data. Consequently, the reliability of the distance field, log maps, and hence the principal geodesics displayed in Figures 2–5 is unverified. This is the single most load-bearing issue because every downstream step depends on g.
[Section 7.1, Figure 2, Eq. (3.2)] The 'semi-analytical expression' for the laminar wake is an ellipse fitted to the tangent-space coordinates obtained by SVD of the encoded data; its parameters (center, amplitude, angle) are fitted values. The paper claims this expression describes 'the solution of Navier-Stokes' (Abstract) and 'a semi-analytical solution of the Navier-Stokes equations' (Section 8). This is circular and overstated: the expression is a parametric curve in the learned latent space, not a solution or invariant manifold of the governing equations. Please clarify the precise sense in which this is 'semi-analytical' and how the fit differs from a regression.
[Section 7.2, Figure 5] The claim that the leading Kolmogorov mode is 'a physical realization of the governing equations' is not supported by quantitative evidence. There is no comparison of the geodesic mode with POD in terms of reconstruction error, captured energy, or projection error; no error bars on the mode trajectory; and no check that the decoded mode lies on the attractor (e.g., by evolving it under the Navier-Stokes equations). Without such metrics, the 'physical' characterization is a visual judgment.
[Section 6, Eqs. (6.2)–(6.3)] The log-map computation relies on numerical approximations of the distance function and its gradient. The paper does not report the error of the final geodesic (e.g., endpoint mismatch or residual of the geodesic equation) for either test case, nor how the multiple-shooting refinement improves the initial interpolated trajectory. Since the tangent-space SVD and the resulting modes inherit errors from these approximations, it would be important to quantify these errors, especially for the Kolmogorov case where the manifold is less regular.

minor comments (6)

[Section 5] The Eikonal solver is described only by referring to previous work (Kelshaw & Magri 2024); please state which components are novel in this paper and provide enough detail for reproducibility, or clearly delineate the contributions.
[Figures 3 and 5] The color scales of the vorticity snapshots are not labeled; please add color bars and state the quantity and units being plotted.
[Section 7.1] The meaning of 'mean residual of 1.734 × 10^{-3}' for the distance function should be defined (e.g., residual of the Eikonal equation or mean absolute error relative to a reference).
[Section 3] The sentence 'we propose the proper latent decomposition (PLD) (Magri & Doan 2022)' reads as if the method was proposed in the cited reference; please rephrase to clarify the present contribution.
[Section 7.2] The statement 'distinct clustering, amenable to singular value decomposition' is vague; state how clustering is measured or why it implies suitability for SVD.
[Eq. (6.2)] The notation dg|b(a) is confusing; consider writing dg(b,a) or dg(b)(a) and explaining the restriction of the distance function.

Circularity Check

2 steps flagged · score 6.0 of 10

The laminar 'semi-analytical solution' and the Kolmogorov 'dominant physical mode' are fitted summaries of the training data in the learned tangent space, so the headline physical results reduce to the data by construction.

fitted input called prediction [Section 7.1 (Laminar wake; Figure 2) and Section 8 (Conclusions)]
"Because of the structure of the data in the tangent space, we can fit a reduced-order model in the tangent space in the form of an ellipse, describing the entire flow field with a single variable, θ. ... ˜zk(θ) = [0.122, −0.0138] + 2.435 [cos(2.879), sin(2.879)] cos(θ) + 0.296 [− sin(2.879), cos(2.879)] sin(θ) ... we are able to obtain principal geodesics, which describe the wake dynamics as well as produce a semi-analytical solution of the Navier-Stokes equations by leveraging the geometry of the attractor."

The elliptic formula is obtained by fitting the encoded training snapshots in the tangent space at the Fréchet mean; the parameters are the sample mean, amplitude, and phase of the SVD of the same data. The 'semi-analytical solution of the Navier-Stokes equations' is exactly this fitted model, so the claimed solution is the input data summarized in new coordinates. No independent derivation from Navier-Stokes or validation on held-out states is provided, so the result is forced by construction.
fitted input called prediction [Section 7.2 (Kolmogorov flow; Figures 4-5) and Section 8 (Conclusions)]
"we observe a distinct clustering, amenable to singular value decomposition. Upon obtaining an orthonormal basis, we re-map the scaled basis vectors to the manifold. ... The trajectory of the leading mode is decoded... We observe a transition from the Fréchet mean, which resembles the forcing term in the Kolmogorov flow, to a snapshot of the flow. ... we are able to identify a dominant mode, which, in contrast to POD, is a physical realization of the governing equations."

The dominant mode is the first right-singular vector of the tangent-space projections of the training snapshots, so it is by construction the direction of maximum variance in the training data under the learned metric. The claim that this mode is a 'physical realization of the governing equations' interprets the fitted SVD direction; it is not obtained from the Navier-Stokes equations nor checked against held-out turbulent states. The physical-looking structure is therefore a property assigned to a data-derived mode, not an independent result.

full rationale

The core PLD pipeline (autoencoder-inferred manifold, metric-constrained Eikonal distance, Fréchet mean, log map, tangent-space SVD, exp map) is a well-defined data-driven procedure and is not itself circular; the citations to Magri & Doan (2022) and Kelshaw & Magri (2024) are method citations that do not by themselves force the reported modes. However, the paper's two headline physical results are presented as discoveries: a 'semi-analytical solution of the Navier-Stokes equations' for the laminar wake and a 'physical realization of the governing equations' for Kolmogorov flow. Both are, by the paper's own description, fits to or SVDs of the same training data in the learned tangent space, with no held-out validation and no derivation from the governing equations. The Section 7.3 conditioning caveat is a numerical-reliability concern rather than circularity, but it reinforces that the computed geodesics depend on the learned metric without independent verification. Overall, the method has independent algorithmic content, but the strongest claims reduce to fitted summaries of the input data, giving partial circularity.

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

PLD introduces no new physical entities. It relies on a set of domain assumptions about the latent manifold and learned metric, plus fitted parameters for the demonstration cases. The central free parameters are the latent dimension and the coefficients of the elliptic reduced-order model, which are computed from the data rather than derived from first principles.

free parameters (4)

Latent dimension dim M = 2
Chosen for visualization in both test cases; not derived from data or error criteria.
Frechet mean (ellipse center) = [0.122, -0.0138]
Computed by gradient descent on the learned distance function; part of the fitted reduced-order model in Section 7.1.
Principal geodesic amplitudes and angle = 2.435, 0.296, 2.879
From SVD of tangent-space data, these define the ellipse axes and orientation in the semi-analytical expression.
Autoencoder and Eikonal network training settings = learning rate 3e-4, 5e4 and 1e5 updates, Adam
Standard training choices; needed for reproduction but not central to the method.

assumptions (5)

domain assumption The flow data lie on (or near) a low-dimensional Riemannian manifold embedded in the autoencoder latent space.
Used throughout, e.g., Section 3 assumes data Z is a subset of a Riemannian manifold M; for Kolmogorov flow a true low-dimensional manifold is not established, and dim M is fixed to 2.
domain assumption The decoder induces a valid Riemannian metric on the latent space, and the Eikonal solver yields an accurate approximation of the geodesic distance.
Sections 5 and 7.3 rely on the learned metric being amenable to distance computation; Section 7.3 concedes condition-number problems.
domain assumption The log map obtained by integrating the gradient of the learned distance function (Eq. 6.2) converges to the length-minimizing geodesic.
Section 6 proposes this iterative procedure without convergence guarantees.
standard math Singular value decomposition of tangent-space data yields energy-ordered modes, and mapping them back via the exp map yields principal geodesics.
This is the standard SVD/PGA construction, assumed in Section 3.
domain assumption Neural network training (autoencoder and distance network) achieves a good local optimum with the stated hyperparameters.
No training curves or multiple seeds are reported; results depend on this assumption.

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Cite this review

Pith. "Pith review of Proper Latent Decomposition." pith.science (2026). https://pith.science/paper/6ISXZO37

@misc{pith2026241200785,
  author       = {Pith},
  title        = {Pith review of: Proper Latent Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ISXZO37}},
  note         = {Machine review of arXiv:2412.00785}
}

read the original abstract

In this paper, we introduce the proper latent decomposition (PLD) as a generalization of the proper orthogonal decomposition (POD) on manifolds. PLD is a nonlinear reduced-order modeling technique for compressing high-dimensional data into nonlinear coordinates. First, we compute a reduced set of intrinsic coordinates (latent space) to accurately describe a flow with fewer degrees of freedom than the numerical discretization. The latent space, which is geometrically a manifold, is inferred by an autoencoder. Second, we leverage tools from differential geometry to develop numerical methods for operating directly on the latent space; namely, a metric-constrained Eikonal solver for distance computations. With this proposed numerical framework, we propose an algorithm to perform PLD on the manifold. Third, we demonstrate results for a laminar flow case and the turbulent Kolmogorov flow. For the laminar flow case, we are able to identify a semi-analytical expression for the solution of Navier-Stokes; in the Kolmogorov flow case, we are able to identify a dominant mode that exhibits physical structures, which are compared with POD. This work opens opportunities for analyzing autoencoders and latent spaces, nonlinear reduced-order modeling and scientific insights into the structure of high-dimensional data.

Figures

Figures reproduced from arXiv: 2412.00785 by the authors.

**Figure 1.** Overview of proper latent decomposition. Given a manifold M (one-dimensional here for pictorial purposes) and data Z ⊂ M (red dots), we wish to find modes, or principal geodesics, that best describe this data. This decomposition comprises three stages: (i) a mean µ ∈ M is computed on the manifold; (ii) data is mapped to the tangent space centered at the mean, TµM, and singular value decomposition is performed to obt… view at source ↗

**Figure 2.** Proper latent decomposition of the laminar flow. Left and right panels show the manifold and the tangent space to the mean respectively. Encoded snapshots of vorticity are depicted as red points, and the resulting geodesic modes are shown in black. An ellipse has been fit to the data in the tangent space, and the resulting trajectory is shown in purple in both panels. The left panel displays contours of geodesic dis… view at source ↗

**Figure 3.** Leading principal geodesic mode for the laminar wake. Snapshots are visualized along the trajectory of the principal geodesic, showing the transition from the Fr´echet mean, to a physical snapshot of the laminar wake. 60 40 20 0 20 40 60 z0 60 40 20 0 20 40 60 z1 Fr'echet mean 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 z0 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 z1 Encoded samples Principal geodesic modes 0 20… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Proper latent decomposition of the Kolmogorov flow. Left and right panels show the manifold and the tangent space to the mean respectively. Encoded snapshots of vorticity are depicted as white points on the manifold, with the colormap showing the distance from the Fr´e…

**Figure 5.** Figure 5: Leading principal geodesic mode for the Kolmogorov flow. Snapshots are visualized along the trajectory of the principal geodesic, showing the transition from the Fr´echet mean to a physical snapshot of turbulence. 150 100 50 0 50 100 150 150 100 50 0 50 100 150 0 200 4…

**Figure 6.** Figure 6: Magnification factor of a prototypical autoencoder trained on the Kolmogorov flow dataset. The left panel demonstrates the log MF across the latent space; the center panel depicts a histogram of the magnification factor, demonstrating a spike at the tail of the distrib…

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ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...
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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...
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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 '...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

[3] [3]

2000 An introduction to the proper orthogonal decomposition

Chatterjee, A. 2000 An introduction to the proper orthogonal decomposition. Curr. Sci. India\/ 78, pp. 808--817

work page 2000

[4] [4]

Chen, R. T. Q., Rubanova, Y., Bettencourt, J. & Duvenaud, D. 2018 Neural ordinary differential equations. In Adv. Neural Inf. Process. Syst. , Vol. 31 (eds. S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi & R. Garnett), pp. 6572--6583. NeurIPS

work page 2018

[5] [5]

Eivazi, H., Veisi, H., Naderi, M. H. & Esfahanian, V. 2020 Deep neural networks for nonlinear model order reduction of unsteady flows. Phys. Fluids\/ 32, 105104

work page 2020

[6] [6]

& Fukagata, K

Fukami, K., Nakamura, T. & Fukagata, K. 2020 Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data. Phys. Fluids\/ 32, 095110

work page 2020

[7] [7]

& Pennec, X

Guigui, N., Miolane, N. & Pennec, X. 2023 Introduction to Riemannian geometry and geometric statistics: From basic theory to implementation with Geomstats . Found. Trends Mach. Learn.\/ 16 , pp. 329--493

work page 2023

[8] [8]

& White, H

Hornik, K., Stinchcombe, M. & White, H. 1989 Multilayer feedforward networks are universal approximators. Neural Netw.\/ 2 , pp. 359--366

work page 1989