{"id":"8f8e1048-df10-46c3-9295-1942a54941e6","arxiv_id":"2504.20288","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"Image interpolation paths are computed as shortest curves under a metric defined by the Jacobian of the diffusion score function.","lead":"A new way to smoothly change one image into another using the mathematical structure hidden inside image-generation models, called diffusion models. If it works, it gives these models a natural sense of geometry for editing and morphing images.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proposed metric G_xt = J^T J is likely degenerate in the manifold-tangent directions that the method must use; Eq. (3) is then ill-posed and the reported geodesic depends on the un-reported regularizer rather than on a well-defined Riemannian metric.","rationale":"Reader's weakest_assumption is the same one I would put first. Section 3 explicitly conditions the validity of Eq. (1) on non-degeneracy of J_xt ('As long as J_xt is non-degenerate'), and never returns to it. The interpretation paragraph then states that small eigenvalues of J_xt correspond to tangent directions of the data manifold, citing work whose entire point is that these eigenvalues are near zero. Squaring them in G = J^T J makes the metric singular or numerically singular in the directions that the geodesic is meant to exploit. As a result, the length functional in Eq. (2) assigns near-zero cost to arbitrary tangent perturbations of the path; Eq. (3) is non-convex and degenerate; and the only things keeping the optimization well-posed are the variance regularizer (with an un-reported λ) and the implicit regularization of Adam with 5,000 iterations. That means Figure 1 and Table 1 do not demonstrate a Riemannian-geometry effect; they demonstrate a particular regularized optimization trajectory. I agree with the reader's verdict: the theoretical foundation is unsound as stated, and the empirical evidence is too thin to rescue it. I would not change the verdict (REJECT): the central claim is unestablished, although a regularized variant (e.g., J^T J + εI with reported ε) could be a legitimate method if tested properly.","tokens_in":11403,"tokens_out":6039,"duration_ms":67339,"concrete_test":"Using one Stable Diffusion pair from Figure 1, reconstruct the endpoints x_0^(0), x_0^(N) by DDIM inversion up to τ=30, initialize the N-1 intermediate points with Slerp as in the paper, and compute the spectrum of G_xt = J_xt^T J_xt at each intermediate point with a randomized SVD (the Jacobian-vector product is already available). Report the condition number λ_max/λ_min and the fraction of spectral mass in the smallest 10% of eigenvalues. If λ_min is at machine precision or κ > 10^6, re-run the optimization with the metric regularized to G_ε = J^T J + εI for ε ∈ {0, 1e-6, 1e-3, 1} and with the variance regularizer removed (λ=0). If the optimized path changes substantially with ε, the unregularized problem is ill-posed and the results in Table 1 are not attributable to the proposed Riemannian metric.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Paper's central construction in Eq. (1) defines G_xt = J_xt^T J_xt with J_xt = ∇_x s_θ(x_t,t), and Section 3 acknowledges this is a Riemannian metric only 'as long as J_xt is non-degenerate'; it is never verified. The paper's own interpretation immediately undercuts the assumption: it states that directions with small eigenvalues of J_xt correspond to movement parallel to the data manifold, citing the spectral-gap analyses of Stanczuk et al. (2024) and Ventura et al. (2025). Those analyses show that, on or near a low-dimensional manifold, the score Jacobian has a cluster of near-zero eigenvalues in tangent directions. For G = J^T J, those eigenvalues are squared, so the metric is near-singular or singular precisely along the directions the geodesic is intended to follow. If G is singular, l(s_i) = sqrt(v_i^T G v_i) does not penalize tangent motion, the minimization in Eq. (3) is degenerate (non-unique minimizers, infinite families of zero-cost tangent perturbations), and the reported path is determined by the auxiliary variance regularizer with un-reported λ and by the Adam optimizer, not by the Riemannian structure. The problem is not merely a missing theoretical nicety: the claimed manifold-following property is supposed to be produced by the metric, but the metric is least reliable in exactly those directions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to treat the data space of a pretrained diffusion model as a Riemannian manifold by defining the metric tensor as G_xt = J_xt^T J_xt, where J_xt is the Jacobian of the score function s_theta(xt, t). The authors then compute the length-minimizing geodesic between two DDIM-inverted noisy samples at time tau, decode the path with DDIM, and compare the resulting interpolations against Lerp, Slerp, NAO, and NoiseDiffusion on a small set of Stable Diffusion prompts and on MNIST. They report qualitative improvements in CLIP-IQA realism, noisiness, and prompt fidelity, while noting that NAO and NoiseDiffusion suffer from reconstruction errors.","tokens_in":11728,"tokens_out":6003,"duration_ms":66183,"significance":"The idea of using the score function to define a geometric structure on diffusion-model data spaces is interesting and connects to recent work on the spectral properties of score Jacobians and manifold learning. If the metric were well-defined and the evaluation were convincing, the framework could offer a principled interpolation method that uses the learned data manifold without retraining. However, the central mathematical premise is not established: the paper itself identifies the regime in which the score Jacobian is near-singular along data-manifold tangent directions, and that is precisely the regime in which the proposed metric degenerates. The experimental evidence is also thin, consisting of four Stable Diffusion examples without statistical analysis. As a result, the paper does not currently support its central claims.","major_comments":[{"comment":"The proposed metric is not a valid Riemannian metric in the intended regime. The paper defines G_xt = J_xt^T J_xt and concedes that it is positive definite only when J_xt is non-degenerate. It then states, citing Stanczuk et al. (2024) and Ventura et al. (2025), that directions with small eigenvalues of J_xt are tangential to the data manifold. Since the data distribution is concentrated near a low-dimensional manifold, J_xt is near-singular in those tangent directions, so G_xt is near-singular along exactly the directions an interpolation path must traverse. The length functional in Eq. (2) then assigns near-zero cost to arbitrary tangent perturbations, and the minimization in Eq. (3) is degenerate; the variance regularizer with un-reported lambda is what actually selects a path. The paper provides no eigenvalue spectra, condition numbers, or regularized variant, so the connection between the reported paths and the claimed Riemannian structure is unsupported.","section":"Section 3, Eq. (1) and \"Interpretation\""},{"comment":"The empirical support for the central claim is limited to four Stable Diffusion examples with no error bars or significance tests. The CLIP-IQA differences between Slerp and the proposed method (Reality 0.704 vs 0.716, Noisiness 0.765 vs 0.818, Fidelity 0.784 vs 0.810) are small and could easily be within run-to-run or image-to-image variability. The proposed method's reconstruction errors are exactly identical to Lerp and Slerp, which is expected because all three use the same DDIM-inverted endpoints; those numbers only confirm that the path optimization does not corrupt endpoints and do not validate the interpolation. The MNIST experiment in Appendix C.2 is presented qualitatively and does not provide quantitative support.","section":"Section 4, Table 1"},{"comment":"The optimization procedure is under-specified in a way that is load-bearing. The variance regularization weight lambda is never reported, the initialization details for the Slerp curve are not given, and the dependence of the final path on the Adam learning-rate schedule and the 5,000-iteration budget is not examined. Because the objective in Eq. (3) is degenerate, the output path depends heavily on these undocumented choices; without a sensitivity analysis, the paper cannot claim that the results reflect a geometric property rather than an artifact of the optimizer or the regularizer.","section":"Section 4 (Implementation) and Appendix B.4"}],"minor_comments":[{"comment":"The denominator of the arccosine argument is written as ||x_t^(0)|| ||x_t^(0)||; the second norm should presumably be ||x_t^(1)||.","section":"Appendix B.3, Eq. (16)"},{"comment":"The paper uses G_xt both as a matrix-valued metric and as a tensor field; explicitly writing indices, e.g., g_{ij}(xt), would avoid confusion between the pointwise matrix and the global geometric object.","section":"Section 3"},{"comment":"The Taylor-expansion justification that the geodesic makes the score change as little as possible is only local and assumes small ||v||; with N=10 discretization points the displacements between successive points may be large, so the interpretation is not directly justified.","section":"Section 3, \"Interpretation\""},{"comment":"The CLIP-IQA fidelity evaluation extends the metric beyond its original design by using prompts such as \"A photo of [object]\" versus \"A photo of something that is not [object]\"; the paper should acknowledge that this is a heuristic adaptation rather than a validated use of the metric.","section":"Section 4, Table 1"}],"recommendation":"reject","confidential_remarks":"The central theoretical flaw is fundamental: the proposed metric degenerates precisely in the directions that the interpolation must follow, and the paper offers no regularization or verification to address this. The missing lambda and the very limited evaluation make it difficult to see how the central claims could be supported without substantial reworking. If the authors were to add a regularizer such as J^T J + epsilon I, they would need to report epsilon, show that the path is stable across a range of epsilon, and explain what geometric meaning the regularized metric has; even then, the paper would need a much more extensive evaluation. This is a workshop-level submission whose core premise is not yet established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nShort version: the paper has a real new idea — define a Riemannian metric on diffusion-model noise space from the score Jacobian, G_xt = J_xt^T J_xt — but the central assumption that this metric is non-degenerate is left unverified, and the paper's own interpretation suggests the opposite. The experiments are too thin to compensate.\n\nWhat is new: prior Riemannian-geometry work on generative models pulls back a metric through a latent space. Here the metric is defined directly from the score function, which I don't think appears in the cited literature. The motivation is clean, and the implementation (Jacobian-vector products, finite-difference velocities, Adam optimization of a discretized length) is reasonable for a workshop paper. The authors also correctly connect the spectral-gap literature (Stanczuk et al., Ventura et al.) to their eigenvalue interpretation.\n\nThe problem sits right there. The paper says directions with small J eigenvalues are tangential to the data manifold. That means the metric is near-singular in the very directions a manifold-following geodesic must move in. At the noise levels used (τ=30 for Stable Diffusion) the distribution is smooth, so J is typically full rank, but the tangent-direction eigenvalues can be orders of magnitude smaller than the normal ones, making the metric extremely ill-conditioned. The variance regularizer is sensible engineering, but λ is unreported, no code is given, and there is no conditioning analysis. So a reader cannot tell how much of the optimized path is determined by the Riemannian structure and how much by the regularizer.\n\nThe empirical support is a proof-of-concept, not a validation. Four Stable Diffusion prompts, with mean errors and no error bars, qualitative curves, and the MNIST experiments have no quantitative results. The NAO comparison uses τ=50 for NAO and τ=30 for everyone else, which stacks the deck against NAO on reconstruction error. The interpolations do look plausible, but the paper does not establish that the geodesic under its metric is a reliable or well-defined tool.\n\nThis is not a reject-the-idea situation. The score metric is worth serious attention, and the paper deserves reviewer time — a referee could ask for the missing λ, a conditioning analysis, and fairer baselines. As it stands, though, the central theoretical assumption is not justified and the evaluation is not persuasive.\n\nRecommendation: send to peer review, expect major revision.","headline":"A genuinely new score-Jacobian metric idea undercut by an unverified non-degeneracy assumption and too-thin, unreproducible experiments.","tokens_in":12211,"tokens_out":4585,"would_cite":false,"duration_ms":50863,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A diffusion model's score Jacobian defines a Riemannian metric whose geodesics produce realistic, low-noise image interpolations.","keywords":["image interpolation","diffusion models","Riemannian metric","score function","geodesic","manifold hypothesis","DDIM inversion","Stable Diffusion"],"falsifier":"Compute the smallest singular value of $\\nabla_{\\mathbf{x}_t} s_\\theta(\\mathbf{x}_t,t)$ at many points on one of the reported Stable Diffusion geodesic paths. If it rounds to zero in any manifold-tangent direction, the unregularized length being minimized is degenerate, and the smooth midpoint images would have to be attributed to the variance regularizer and the DDIM decoder rather than to the Riemannian metric itself.","tokens_in":11168,"feed_emoji":"🖼️","tokens_out":9754,"duration_ms":91407,"temperature":0.7,"pith_summary":"This paper claims that interpolation between two images can be treated as a geodesic on the data manifold that a pre-trained diffusion model has learned, rather than as a straight line in pixel or noise space. The metric is built entirely from the score function: the length of a small movement is the amount by which the score changes, so the cheapest paths are the ones that stay in regions where the score is nearly constant. Running those geodesics in a mildly noisy space and then decoding with DDIM yields transitions that the paper's CLIP-IQA and reconstruction comparisons say are more realistic, less noisy, and more faithful to the prompt than Lerp, Slerp, NAO, and NoiseDiffusion. If this holds, diffusion models get a practical geometric tool for image interpolation and editing without retraining or a dedicated latent space.","feed_headline":"Score-based geodesics beat prior diffusion interpolation methods","feed_subtitle":"A metric from the diffusion score's Jacobian keeps interpolated frames on the data manifold, cutting blur and artifacts.","key_machinery":"The load-bearing object is the score-Jacobian metric $G_{\\mathbf{x}_t} = (\\nabla_{\\mathbf{x}_t} s_\\theta(\\mathbf{x}_t,t))^{\\top} \\nabla_{\\mathbf{x}_t} s_\\theta(\\mathbf{x}_t,t)$, a matrix that converts the score model's local sensitivity into a notion of distance. In this metric, a small step is cheap when the score is locally flat, so geodesics are paths that keep the score, and therefore the semantic content, as stable as possible. The supporting mechanism is numerical: finite-difference velocities, a trapezoidal length estimate, Adam optimization of the intermediate points, a variance term that keeps the Euclidean speed even, and DDIM inversion and decoding to carry the path through a less rugged time-$\\tau$ space.","core_discovery":"The paper's central claim is that the score function of a diffusion model supplies a Riemannian metric on the data space: at each noisy state $\\mathbf{x}_t$, the metric tensor is $G_{\\mathbf{x}_t} = J_{\\mathbf{x}_t}^{\\top} J_{\\mathbf{x}_t}$, where $J_{\\mathbf{x}_t} = \\nabla_{\\mathbf{x}_t} s_\\theta(\\mathbf{x}_t,t)$ is the Jacobian of the trained score. Under this metric, the length of a velocity is the Euclidean norm of the score perturbation caused by that velocity, so the shortest curve between two endpoints is the path along which the score changes least. The paper computes that path in the time-$\\tau$ noisy space, initializes it with Slerp, optimizes the discrete curve length with a variance regularizer, and decodes the result with DDIM. Experiments on Stable Diffusion and MNIST are reported to show that this geodesic interpolation produces more realistic, less noisy, and more prompt-faithful intermediate images than Lerp, Slerp, NAO, and NoiseDiffusion, while avoiding the reconstruction errors of NAO and NoiseDiffusion.","pith_inferences":["A testable extension the authors do not run is to remove the variance regularizer: if the true geodesic still produces even, meaningful transitions, the metric is doing the work; if the path collapses or concentrates near the endpoints, the regularizer is the effective engine.","A neighbouring application not explored in the paper is to repurpose the score-Jacobian length as a perceptual distance or similarity measure for image retrieval and morphing.","Because $J^{\\top} J$ is singular on low-dimensional manifolds, a regularized version $J^{\\top} J + \\varepsilon I$ would likely make the optimization more stable; whether it changes the visual output is a cheap, concrete experiment.","Another natural stress test is to replace Stable Diffusion with a different pre-trained diffusion model and check whether the reported CLIP-IQA gains persist, which would show the metric is a general property of score functions rather than an artifact of one architecture."],"forward_implications":["Any pre-trained diffusion model can be given a geometric interpolation routine without retraining or learning a separate latent space, because the metric is computed from the score the model already predicts.","Because the geodesic length is the total score change along a path, the same construction defines a distance between images, opening a route to geometry-aware retrieval or editing.","The method preserves endpoint fidelity: reconstructions near the endpoints remain close to the originals under MSE, LPIPS, and DreamSim, whereas NAO and NoiseDiffusion can seriously degrade them.","The score-Jacobian link to the Hessian of the log-density ties the smoothness of interpolation to the intrinsic dimensionality of the data manifold, so the geodesic should trace perceptually meaningful transitions wherever the trained score is well calibrated."],"supporting_citations":[{"why":"Supplies the Stable Diffusion backbone used in all text-to-image experiments.","marker":"Rombach et al., 2022"},{"why":"Defines the diffusion process whose score is used and provides the Lerp interpolation baseline.","marker":"Ho et al., 2020"},{"why":"Provides DDIM, used for deterministic inversion into the noisy space and as the reverse decoder back to images.","marker":"Song et al., 2021a"},{"why":"Provides the Slerp baseline whose path initializes the geodesic optimization.","marker":"Shoemake, 1985"},{"why":"Defines the NAO baseline that the method must beat on fidelity and noisiness.","marker":"Samuel et al., 2023"},{"why":"Defines the NoiseDiffusion baseline; the comparison highlights the proposed method's smaller reconstruction error.","marker":"Zheng et al., 2024"},{"why":"Prior work linking eigenvalues of the score Jacobian to the data manifold's intrinsic dimension, grounding the metric's interpretation.","marker":"Stanczuk et al., 2024"},{"why":"DDIM inversion technique used to map images into the time-tau space for interpolation.","marker":"Mokady et al., 2023"},{"why":"Supplies the CLIP-IQA metric used to evaluate realism, noisiness, and prompt fidelity.","marker":"Wang et al., 2023"}],"fun_headline_variants":["Score metric geodesics beat prior interpolation","Riemannian diffusion: score-defined interpolation paths","Cleaner diffusion interpolation via score geometry","Score Jacobian metric yields sharper interpolation","Diffusion score metric: realistic image interpolation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction assumes the score Jacobian is non-degenerate everywhere along the interpolation path; on a low-dimensional data manifold, directions along the manifold have near-zero score gradient, so the metric can become singular and the geodesic length is only well-defined after some regularization.","fun_headline_variants_meta":{"raw":{"variants":["Score metric geodesics beat prior interpolation","Riemannian diffusion: score-defined interpolation paths","Cleaner diffusion interpolation via score geometry","Score Jacobian metric yields sharper interpolation","Diffusion score metric: realistic image interpolation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1364,"prompt_tokens":861,"completion_tokens":503,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":477,"completion_tokens_details":{"reasoning_tokens":439}},"tokens_in":477,"tokens_out":503,"duration_ms":4941,"temperature":1.0,"reasoning_tokens":439,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:32:38.273085+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the smallest singular value of $\\nabla_{\\mathbf{x}_t} s_\\theta(\\mathbf{x}_t,t)$ at many points on one of the reported Stable Diffusion geodesic paths. If it rounds to zero in any manifold-tangent direction, the unregularized length being minimized is degenerate, and the smooth midpoint images would have to be attributed to the variance regularizer and the DDIM decoder rather than to the Riemannian metric itself.","supporting_citations":[{"cited_title":"High-resolution Image Synthesis with Latent Diffusion Models","cited_arxiv_id":null,"evidence_quote":"Supplies the Stable Diffusion backbone used in all text-to-image experiments."},{"cited_title":"Animating Rotation with Quaternion Curves","cited_arxiv_id":null,"evidence_quote":"Provides the Slerp baseline whose path initializes the geodesic optimization."},{"cited_title":"Norm-guided Latent Space Exploration for Text-to-image Generation","cited_arxiv_id":null,"evidence_quote":"Defines the NAO baseline that the method must beat on fidelity and noisiness."},{"cited_title":"Diffusion Models Encode the Intrinsic Dimension of Data Manifolds","cited_arxiv_id":null,"evidence_quote":"Prior work linking eigenvalues of the score Jacobian to the data manifold's intrinsic dimension, grounding the metric's interpretation."},{"cited_title":"Null-Text Inversion for Editing Real Images using Guided Diffusion Models","cited_arxiv_id":null,"evidence_quote":"DDIM inversion technique used to map images into the time-tau space for interpolation."},{"cited_title":"Chan, and Chen Change Loy","cited_arxiv_id":null,"evidence_quote":"Supplies the CLIP-IQA metric used to evaluate realism, noisiness, and prompt fidelity."}],"review_version":1}