{"id":"cb9c465d-7235-4f8a-8f35-ae097141ce66","arxiv_id":"2506.02254","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A generative sampling method that runs a full-order stochastic differential equation in a Double Diffusion Maps latent space and lifts samples back to the data space via Geometric Harmonics.","lead":"The paper combines two existing manifold-learning tools, Double Diffusion Maps and Geometric Harmonics, into an upgraded version of the Probabilistic Learning on Manifolds framework for generating new samples from a small dataset. The authors demonstrate the approach on Hermite polynomial data and on rotating detonation engine simulations, but provide only visual validation and no comparison to the original method.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The low-data generalization claim rests on GH lifting accuracy for ISDE-generated latent points, which is never quantified and is most fragile precisely in the small-N regime the paper targets.","rationale":"The reader's weakest assumption identifies the functional dependence of discarded eigenvectors as the load-bearing condition, which is a necessary theoretical premise. I agree that this is central, but the more concrete and testable risk is that the GH-based lifting is never validated on the actual out-of-sample latent points produced by the ISDE, and that the claimed low-data regime is not tested at all. The functional-dependence assumption is partially supported by the residual analysis (Figure 6) and the in-sample reconstruction plots (Figure 7), so it is not obviously false; however, even if the discarded eigenvectors are exactly functions of the retained coordinates on the training set, the Nystrom extension used by GH has no guarantee for points far from the training data, and the ISDE can generate such points because its target KDE has unbounded support. This is precisely a low-N failure mode, since small N leads to larger bandwidths and more diffuse sampling. The paper's evaluation is almost entirely visual and lacks any quantitative comparison to original PLoM or standard baselines, so the central claim of improved sample quality and generalization remains unsubstantiated. My concern does not move the verdict: CONDITIONAL is already the right call, because the framework may be sound but requires quantitative validation and reproducibility artifacts. I would add a specific requirement: demonstrate GH lifting accuracy on ISDE-generated latent coordinates and benchmark against original PLoM in genuinely small-N settings.","tokens_in":15384,"tokens_out":8236,"duration_ms":79403,"concrete_test":"On the Hermite D7 dataset, train GH-PLoM for N in {100, 300, 1000, 10000}. For each N, generate 2000 samples and compute: (i) median GH lifting error on held-out true samples (latent coordinates via Nystrom extension, then GH lift, then L2 error vs original); (ii) empirical covariance of generated vs true samples, checking for variance shrinkage; (iii) MMD or energy distance between generated and held-out samples; (iv) the same metrics for original PLoM and a KDE+GH baseline without the ISDE. If GH lifting error is large for generated latent points far from the training cloud, or if GH-PLoM does not beat PLoM at N<=300, the low-data generalization claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that GH-PLoM improves sample quality and generalization in low-data regimes. For this to hold, the Geometric Harmonics inverse map GH^-1 must accurately lift arbitrary latent samples produced by the full-order ISDE back to the ambient space. The paper asserts in Section 4 that discarded diffusion-map eigenvectors are functionally dependent on the retained coordinates and can be reconstructed through the latent harmonics Psi, but this is only demonstrated in-sample (Figure 7) and with visual comparisons (Figures 8, 14-16). No experiment measures the GH reconstruction error on actual out-of-sample latent points generated by the ISDE. Because the ISDE targets a Gaussian KDE with unbounded support, generated latent coordinates can fall outside the training cloud; the Nystrom/GH extension error grows with distance from the training data and is uncontrolled. This problem is exacerbated exactly in the low-N regime the paper claims to enable, where KDE bandwidths are larger and tails are heavier, and where the diffusion-map eigenvectors themselves are noisier. Moreover, the experiments do not test the claimed regime: N=10,000 (Hermite) and N=897 (RDE) are moderate-to-large, there is no comparison against original PLoM or any generative baseline, and no quantitative sample-quality metric (MMD, energy distance, coverage, covariance bias) is reported. The central claim is therefore unsupported by the evidence presented.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes GH-PLoM, an extension of Probabilistic Learning on Manifolds (PLoM) that combines Double Diffusion Maps with Geometric Harmonics (GH) and solves a full-order Itô stochastic differential equation (ISDE) directly in a diffusion-map latent space. The method first embeds high-dimensional data into a low-dimensional manifold using diffusion maps, constructs a latent harmonic basis via a second diffusion-map step, estimates a kernel density on the latent coordinates, generates new latent samples from a full-order ISDE, and lifts them back to the ambient space using the GH inverse map. The authors claim that this hybrid approach retains the geometric expressiveness of PLoM while improving sample quality and generalization in low-data regimes. The paper includes two numerical studies: one based on two-dimensional Hermite polynomial functions (N = 10,000) and one based on detonation-wave simulations in a rotating detonation engine (N = 897). The reported results are primarily visual, showing that generated samples resemble training data in terms of mean fields and representative realizations.","tokens_in":15723,"tokens_out":3277,"duration_ms":31652,"significance":"The conceptual integration of Double Diffusion Maps and Geometric Harmonics into PLoM is a sensible and potentially valuable direction: it addresses a real limitation of the original method, namely the absence of a principled inverse map from the diffusion-map latent space back to the ambient data space. The proposed framework is clearly described, and the two test problems are nontrivial, particularly the rotating detonation engine example with high-dimensional input vectors. If the central claim of improved generalization in low-data regimes were properly supported, this would be a useful contribution to generative modeling on manifolds. However, as presented, the evidence does not substantiate that claim: the experiments do not operate in the small-data regime the paper motivates, no comparison with the original PLoM or other generative baselines is provided, and no quantitative sample-quality or lifting-error metrics are reported. The paper's contribution is therefore currently an interesting algorithmic proposal with preliminary visual demonstrations rather than a validated method.","major_comments":[{"comment":"The paper's central claim, stated in the introduction and conclusion, is that GH-PLoM improves sample quality and generalization in low-data regimes. This claim is not supported by the experiments. The Hermite example uses N = 10,000 samples and the RDE example uses N = 897, both of which are moderate-to-large relative to the intrinsic dimensions of the respective manifolds (roughly 1-2 for Hermite and 3 for RDE). The motivating regime described in Section 3 is when the number of diffusion-map basis vectors approaches N in 'extreme small-data settings', and neither experiment approaches that regime. Moreover, there is no comparison against the original PLoM or any other generative baseline, and no quantitative sample-quality metric (such as MMD, energy distance, coverage, or covariance bias) is reported; Figures 8, 14, 15, and 16 are visual only. To support the central claim, the authors should add experiments with small N (e.g., N comparable to the diffusion-map truncation dimension), compare against the original PLoM and a standard baseline, and report quantitative metrics of sample quality and generalization.","section":"Section 6 and Section 7"},{"comment":"The accuracy of the Geometric Harmonics inverse map GH^{-1} for out-of-sample latent points generated by the ISDE is never quantified, yet it is load-bearing for the method. The only quantitative GH check, Figure 7, evaluates reconstruction on points from the training distribution (train and test splits of the observed data cloud), not on latent coordinates produced by the full-order ISDE. Because the kernel density estimate in Eq. (5) has unbounded support on R^nu, the ISDE can generate latent points that lie outside the convex hull of the training latent coordinates; for such points, the Nyström extension and GH interpolation error is uncontrolled and generally grows with distance from the training data. This is precisely the failure mode that matters in the low-data regime the paper claims to enable, where KDE bandwidths are larger and the diffusion-map eigenvectors are noisier. The authors should measure GH reconstruction error on actual ISDE-generated latent points, for both in-distribution and extrapolation cases, and report how this error scales with N and with distance from the training cloud.","section":"Section 5 and Algorithm 2"},{"comment":"The key assumption underlying Double Diffusion Maps as used here is that the discarded diffusion-map eigenvectors are 'functionally dependent' on the retained non-harmonic coordinates and can therefore be reconstructed through the latent harmonics Psi. This assumption is asserted in the first paragraph of Section 4 without supporting evidence or error analysis. Since every generated ambient sample is obtained by lifting through this reconstructed inverse map, the accuracy of this functional dependence is load-bearing: if it holds only approximately, the approximation error propagates into all generated samples. The manuscript should either provide a theoretical bound on the reconstruction error, quantify it empirically on held-out and ISDE-generated points, or explicitly state the conditions under which the assumption is valid and acknowledge the resulting limitation.","section":"Section 4, first paragraph"}],"minor_comments":[{"comment":"There are several unresolved '[REF]' placeholders in the text (e.g., in the introduction and in the caption of Figure 9). These must be completed before the manuscript can be considered publication-ready.","section":"Throughout"},{"comment":"Equation numbering is inconsistent: Eq. (9) appears twice (once for the mean identity and once for the product-form density), and Eq. (10) is used for both the ISDE and the covariance identity. Please renumber all equations.","section":"Section 2, equations"},{"comment":"Table 2 contains a typo: 'Psi_{0,2)' is missing the closing subscript parenthesis; it should be 'Psi_{(0,2)}'.","section":"Table 2"},{"comment":"The text states that 'the noise has vanished since we are taking the expectation over 100 generated with PLoM realizations'. Averaging over realizations removes stochastic noise by construction, so this observation does not demonstrate that individual generated samples are statistically consistent or noise-free; the authors should clarify what claim is being made.","section":"Section 6.1, Figure 8"},{"comment":"The caption of Figure 16 refers to an 'augmented dataset' while the text describes a 'reconstructed image'; please unify the terminology to avoid confusion between generated samples and reconstructed training samples.","section":"Section 6.2, Figure 16"},{"comment":"When discussing limitations of the original PLoM in small-data settings, the authors should cite the specific prior extensions that already address small-data regimes (e.g., Refs. [26], [35], and [36]) rather than only the original PLoM paper, so that the claimed novelty relative to those extensions is made explicit.","section":"Section 3, challenges"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents a plausible algorithmic extension, but the evidence gap between the central claim (improved sample quality and generalization in low-data regimes) and the experiments (moderate-to-large N, no baselines, no quantitative metrics, no out-of-sample lifting error analysis) is substantial. I would encourage the editor to invite a major revision that adds small-N experiments, comparisons against original PLoM and a standard generative baseline, and quantitative evaluation of both sample quality and GH lifting error on ISDE-generated points. The paper is within scope for the journal, but the current form is more of a methods proposal than a validated contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the bottom line: this is a plausible, well-written extension of PLoM, but the paper never tests its own central claim. The combination of Double Diffusion Maps and Geometric Harmonics to define the inverse map, with the full-order ISDE run in the latent space, is genuinely new as far as I know. The math is laid out clearly and the RDE example is the kind of complex problem the method is aimed at. I don't see any circular reasoning; the machinery is imported from published work and the authors are transparent about that.\n\nWhat the paper does well is the exposition. The PLoM review is thorough, the Double Diffusion Maps and GH steps are explained carefully, and the idea of using GH for lifting is natural. The extreme-sample reconstruction in the RDE case is a nice qualitative check.\n\nThe soft spots are real and load-bearing. The introduction promises 'improving sample quality and generalization in low-data regimes,' but neither experiment is in a low-data regime: Hermite uses N=10,000 and RDE uses N=897. There's no comparison to original PLoM or any other generative baseline, and no quantitative sample-quality metric anywhere—no MMD, energy distance, or even covariance bias. The GH inverse map's accuracy on actual ISDE-generated latent points is never measured. The stress-test note is correct: the functional dependence of discarded eigenvectors is asserted but only demonstrated in-sample, and the KDE's unbounded support means latent samples can fall outside the training cloud where Nyström error is uncontrolled. That is exactly the regime the paper claims to address. There are also a few unresolved placeholders ([REF] in the introduction and Figure 9 caption) that suggest the manuscript isn't fully polished.\n\nWho is this for? People actively working on PLoM or manifold-based generative models will want to read it. It deserves a serious referee because the synthesis is new and the authors are credible, but the referee should insist on small-N experiments, a direct PLoM comparison, and out-of-sample GH error quantification. I'd send it out, with the expectation of major revision.","headline":"The synthesis is new and clean, but the paper's headline claim about low-data generalization is not tested; it needs quantitative evidence and baselines before it can be believed.","tokens_in":16238,"tokens_out":2808,"would_cite":false,"duration_ms":25004,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62R30","60H10","65C30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that extending Probabilistic Learning on Manifolds with Double Diffusion Maps and Geometric Harmonics, and solving a full-order stochastic differential equation in the latent space, produces statistically consistent…","keywords":["generative modeling","Probabilistic Learning on Manifolds","Double Diffusion Maps","Geometric Harmonics","latent space sampling","stochastic differential equations","manifold learning","small-data uncertainty quantification"],"falsifier":"Take a dataset whose manifold is intentionally chosen so that a discarded diffusion coordinate is not a function of the retained ones, such as an independent extra mode with small variance, and compute the leave-one-out Geometric Harmonics reconstruction error for that coordinate from the retained latent coordinates; if the normalized error does not shrink toward zero as the sample size grows, the functional-dependence premise fails, and generated samples will inherit that error. A second check is to sample from the full-order ISDE in the latent space, lift the samples, and compare the empirical distribution of the lifted samples against the training distribution on a held-out test set.","tokens_in":15194,"feed_emoji":"🌀","tokens_out":9488,"duration_ms":80460,"temperature":0.7,"pith_summary":"The paper proposes an extension of Probabilistic Learning on Manifolds (PLoM) aimed at generating statistically consistent samples of a random vector whose distribution concentrates on a low-dimensional manifold, using only a small set of observed data. Its central claim is that combining Double Diffusion Maps with Geometric Harmonics and solving a full-order stochastic differential equation in the latent space, rather than a reduced-order one, preserves the geometric structure of the data while improving sample quality and generalization when the number of data points is small. The paper further claims that the new method supplies a principled inverse map from latent coordinates back to the original high-dimensional space, something the original formulation lacked. If these claims are right, manifold-based generative modeling becomes practical for scarce scientific datasets, including expensive simulation ensembles.","feed_headline":"Latent map lifts samples back to the data manifold faithfully","feed_subtitle":"Double Diffusion Maps plus Geometric Harmonics give generators a smooth way back from latent space, even with few data.","key_machinery":"The central object is the Double Diffusion Maps latent harmonics basis $\\Psi$, obtained by running a second diffusion map on the leading non-harmonic diffusion coordinates $g_\\alpha$ with the kernel $k^\\star_\\epsilon(g_\\alpha^{(i)},g_\\alpha^{(j)}) = \\exp(-\\|g_\\alpha^{(i)}-g_\\alpha^{(j)}\\|^2/2\\epsilon^2)$. This basis supports Geometric Harmonics, a spectral interpolation scheme that projects any function $h$ defined on the data onto the latent harmonics, $P_\\delta h = \\sum_{j=1}^m \\langle h,\\psi_j\\rangle \\psi_j$, and evaluates it at a new latent point through the out-of-sample extension $\\Psi_j(g^{\\mathrm{new}}_\\alpha)=\\sigma_j^{-1}\\sum_{i=1}^m k^\\star_\\epsilon(g^{\\mathrm{new}}_\\alpha,g^{(i)}_\\alpha)\\psi_j(g^{(i)}_\\alpha)$. The reconstructed map $GH^{-1}:[g_m]\\to[\\psi]\\to[x_d]$ carries generated latent samples back to the ambient space, so the full-order stochastic differential equation can be solved in the low-dimensional latent space while the generated samples still land on the learned manifold.","core_discovery":"On the paper's own terms, the central discovery is that the small-data failure of PLoM, where the diffusion-map basis dimension approaches the sample size and generalization is lost, can be removed by learning the inverse map explicitly. The method first applies Diffusion Maps to find the leading non-harmonic coordinates of the data manifold, then applies a second Diffusion Map on those coordinates to build a latent harmonics basis, and uses Geometric Harmonics to extend functions from the latent space back to the ambient space. A full-order ISDE is solved directly in the latent space, with the estimated density of the latent coordinates as its invariant measure, and each generated latent sample is lifted through the Geometric Harmonics interpolant to produce a new high-dimensional realization. The paper reports that this procedure retains the global structure of polynomial test data and reproduces both ensemble-averaged fields and extreme outlier realizations for detonation-wave simulations.","pith_inferences":["Because the latent harmonics basis is constructed from the data itself, the method offers a nonparametric alternative to autoencoder-based generators for manifold-supported distributions, with no neural-network training; this is a natural consequence of the construction that the paper does not spell out.","The Geometric Harmonics reconstruction error of the discarded coordinates could serve as a diagnostic for choosing the latent dimension, complementing the residual-based criterion used in the paper.","One could test the low-data advantage directly by running the proposed method and the original PLoM on the same datasets while progressively reducing the sample size, and comparing the distribution of lifted samples against a held-out reference; the paper's two examples suggest but do not report such a comparison.","The same latent-space SDE plus Geometric Harmonics lifting could be adapted to conditional generation by augmenting the latent variables with conditioning parameters, an extension the authors leave for future work."],"forward_implications":["The method supplies a smooth inverse map from latent coordinates to the observation space, so generated samples can be visualized and evaluated directly as physical fields rather than only as abstract reduced coordinates.","Solving the full-order ISDE in the latent space preserves the full dynamical complexity of the system instead of projecting the dynamics onto a truncated diffusion basis.","In low-data settings the method should avoid the overfitting plateau where the diffusion-map basis dimension approaches the sample size, because the latent space stays low-dimensional and discarded directions are reconstructed rather than discarded.","The framework applies to datasets whose features are uncorrelated but still lie on a low-dimensional manifold, where linear reduction such as principal component analysis fails.","The reported reproduction of both ensemble-averaged behavior and extreme outlier realizations indicates the method can support uncertainty quantification on costly simulations."],"supporting_citations":[{"why":"Defines the original PLoM framework that this paper extends, supplying the kernel density estimation, diffusion-map, and reduced-order ISDE components.","marker":"[24]"},{"why":"Introduces Diffusion Maps, the nonlinear manifold-learning method used to extract the latent coordinates.","marker":"[31]"},{"why":"Introduces Double Diffusion Maps and latent harmonics, the central construction the proposed method builds on.","marker":"[32]"},{"why":"Introduces Geometric Harmonics, the spectral interpolation scheme used for lifting latent samples to the ambient space.","marker":"[33]"},{"why":"Provides the leave-one-out local-linear-regression residual that identifies non-harmonic eigenvectors and sets the latent dimension.","marker":"[34]"},{"why":"Supplies the rotating detonation engine simulation data used in the second numerical demonstration.","marker":"[40]"}],"fun_headline_variants":["Double Diffusion Maps beat small-data PLoM overfitting","Multiscale maps restore generative learning on few samples","Latent-space ISDE with geometric harmonics lifts samples","Small-data generative model gains via double diffusion maps","Double diffusion maps enable probabilistic manifold learning"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes that every diffusion-map direction that is discarded is exactly determined by the retained coordinates, so that Geometric Harmonics can reconstruct those discarded directions from any new latent point; if that functional dependence is only approximate, the lifting step injects error into every generated sample.","fun_headline_variants_meta":{"raw":{"variants":["Double Diffusion Maps beat small-data PLoM overfitting","Multiscale maps restore generative learning on few samples","Latent-space ISDE with geometric harmonics lifts samples","Small-data generative model gains via double diffusion maps","Double diffusion maps enable probabilistic manifold learning"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1285,"prompt_tokens":994,"completion_tokens":291,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":218}},"tokens_in":610,"tokens_out":291,"duration_ms":3113,"temperature":1.0,"reasoning_tokens":218,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:27:51.881314+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a dataset whose manifold is intentionally chosen so that a discarded diffusion coordinate is not a function of the retained ones, such as an independent extra mode with small variance, and compute the leave-one-out Geometric Harmonics reconstruction error for that coordinate from the retained latent coordinates; if the normalized error does not shrink toward zero as the sample size grows, the functional-dependence premise fails, and generated samples will inherit that error. A second check is to sample from the full-order ISDE in the latent space, lift the samples, and compare the empirical distribution of the lifted samples against the training distribution on a held-out test set.","supporting_citations":[{"cited_title":"Data-driven probability concentration and sampling on manifold.Journal of Computational Physics, 321:242–258, 2016","cited_arxiv_id":null,"evidence_quote":"Defines the original PLoM framework that this paper extends, supplying the kernel density estimation, diffusion-map, and reduced-order ISDE components."},{"cited_title":"Diffusionmaps","cited_arxiv_id":null,"evidence_quote":"Introduces Diffusion Maps, the nonlinear manifold-learning method used to extract the latent coordinates."},{"cited_title":"Double diffusion maps and their latent harmonics for scientific computations in latent space.Journal of Computational Physics, 485:112072, 2023","cited_arxiv_id":null,"evidence_quote":"Introduces Double Diffusion Maps and latent harmonics, the central construction the proposed method builds on."},{"cited_title":"Geometric harmonics: a novel tool for multiscale out-of- sample extension of empirical functions.Applied and Computational Harmonic Analysis, 21(1):31–52, 2006","cited_arxiv_id":null,"evidence_quote":"Introduces Geometric Harmonics, the spectral interpolation scheme used for lifting latent samples to the ambient space."},{"cited_title":"Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study.Applied and Computational Harmonic Analysis, 44(3):759–773, 2018","cited_arxiv_id":null,"evidence_quote":"Provides the leave-one-out local-linear-regression residual that identifies non-harmonic eigenvectors and sets the latent dimension."},{"cited_title":"Uncertainty Quantification and Flow Dynamics in Rotating Detonation Engines","cited_arxiv_id":"2505.02378","evidence_quote":"Supplies the rotating detonation engine simulation data used in the second numerical demonstration."}],"review_version":1}