{"id":"a2181d9e-84ef-4539-9327-19e4f7e6395e","arxiv_id":"2412.04282","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A dynamic Gaussian rendering method that wraps a third-order time polynomial around an SC-GS style skeleton and reports higher PSNR, though the 'infinite Taylor' framing is a fitted residual network.","lead":"This paper models the motion of 3D Gaussian splats in dynamic video as a third-order Taylor polynomial plus a neural-network residual, and reports state-of-the-art numbers on two public datasets. A generalist reader should look closely because the central mathematical claim, a complete Taylor series without approximation, is not supported by the implementation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'complete Taylor series' claim fails because the MLP/LBS residual in Eq. (11) is not constrained to be a Peano remainder; the decomposition is a tautology, not a Taylor expansion.","rationale":"The reader's weakest assumption identifies the exact load-bearing point: Eq. (11) is vacuous unless H_k satisfies the Peano remainder condition, and the construction in Section 4.4 contains no mechanism that would enforce H_k(t) = o((t - t_tau)^k). I agree with that diagnosis. I also concur with the secondary observations about undefined Taylor coefficients and implausible baseline numbers (e.g., D3DGS at 20-23 dB PSNR on N3DV and SC-GS at 6.29 dB on Cut Roasted Beef are far below published values), but I treat those as supporting evidence rather than a separate load-bearing objection, because without released code one cannot fully adjudicate the experimental pipeline. The paper does include an internal ablation (Table 3) showing that each time-varying component contributes, and the method may be a plausible incremental rendering recipe; that is real, but it is not the advertised novelty. The rejection is driven by an internal inconsistency with Taylor's theorem, not by disagreement with community consensus. The concrete test above would settle the matter directly: it inspects the trained model's residual near the expansion center and checks the defining property of a Peano remainder. If the property is absent, the central claim is unsupported and the paper should not be accepted as presenting a learnable infinite Taylor series.","tokens_in":14818,"tokens_out":6632,"duration_ms":68502,"concrete_test":"For a trained model on any N3DV scene, take a fixed Gaussian point and a time center t_tau on the training grid; set delta = 2^-p * frame_dt for p = 0..8 and evaluate the actual residual H_k(t_tau +/- delta) used by the model, separating the third-order polynomial part from the LBS/MLP residual according to the implementation's forward pass. Compute |H_k(t)| / |t - t_tau|^3 for both signs. If these ratios do not tend to 0 as delta -> 0, then H_k is not a Peano remainder and the 'complete Taylor series without approximation' claim is false. A minimal non-code version of the same check: derive from Section 4.3-4.4 an explicit formula for the coefficients f_p^(k), f_s^(k), f_q^(k); if no such formula exists, the Taylor expansion in Eqs. (12)-(14) has no defined content.","verdict_should_be":"REJECT","load_bearing_attack":"The paper's stated novelty is that Eq. (11) 'constructs a complete Taylor series that estimates the motion model without relying on approximations' (Section 4.2 and the third contribution). For that to be true, H_k(t) must be the Peano remainder of the third-order Taylor polynomial f_k(t), i.e. H_k(t) = o((t - t_tau)^k) as t -> t_tau. Nothing in Section 4.4 imposes this condition. H_k is obtained from an MLP decoder on sparse control points (Eq. 15) and then blended to local Gaussians through Linear Blend Skinning (Eqs. 16-18). The blended offset contains terms like R_j^t(mu_i - p_j) + p_j + Delta d_j^t and a quaternion product; these quantities are learned without any requirement that they vanish, let alone faster than (t - t_tau)^k, when t approaches the expansion center t_tau. The time-dependent opacity RBF in Eq. (7) is unrelated to the position/rotation/scale remainders. Moreover, a third-order polynomial plus one residual is not an 'infinite' Taylor series: no higher-order coefficients, no sequence, and no limiting argument are provided. The coefficients f_p^(k)(t_tau), f_s^(k)(t_tau), f_q^(k)(t_tau) in Eqs. (12)-(14) are never defined, and the orders n, m, l are left unspecified, so f_k is not even a concrete Taylor polynomial. Consequently, Eq. (11) is a tautological decomposition valid for any H_k; it does not derive predictive or interpretive power from Taylor's theorem. The rendering recipe might still work, but the mathematical framing that the paper advertises as its central contribution is unsupported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a dynamic 3D Gaussian Splatting method in which the time evolution of Gaussian position, rotation, and scale is modeled as a \"learnable infinite Taylor series.\" Equation (11) decomposes the spatiotemporal transformation Ti(t) into a polynomial component fk(t), described as a third-order Taylor expansion, and a residual Hk(t), described as a Peano remainder and implemented with an MLP decoder plus linear blend skinning. The paper reports quantitative and qualitative results on the N3DV and Technicolor datasets, claiming state-of-the-art dynamic view rendering performance, and includes ablations of the time-dependent motion, opacity, and remainder components.","tokens_in":15150,"tokens_out":7142,"duration_ms":69952,"significance":"If the central theoretical claim were valid, the method would offer an appealing combination of interpretable polynomial dynamics and flexible learned residuals for dynamic Gaussian splatting, and the reported PSNR, SSIM, and LPIPS numbers would make a strong empirical case. The manuscript contains extensive experiments on two public datasets and a clearly described pipeline. However, the advertised mathematical contribution is not established: the decomposition in Eq. (11) is not a Taylor expansion in any rigorous sense, the term \"infinite Taylor series\" is not realized by the proposed construction, and the baseline tables contain numbers that appear broken or inconsistent. These issues undermine both the novelty claim and the empirical comparison, so the paper in its current form does not support its central conclusions.","major_comments":[{"comment":"The central claim that Ti(t) = fk(t) + Hk(t) \"constructs a complete Taylor series that estimates the motion model without relying on approximations\" is not supported. For Hk(t) to be the Peano remainder of a third-order Taylor polynomial, it must satisfy Hk(t) = o((t - t_tau)^3) as t -> t_tau, and in particular Hk(t_tau) = 0. In Section 4.4, Hk is obtained from an MLP decoder on sparse control points and then blended via linear blend skinning (Eqs. 15-18); no vanishing or locality condition is imposed, and the predicted offsets and rotations can be nonzero at the expansion center. As written, Eq. (11) is a tautological decomposition valid for any function Ti and does not derive from Taylor's theorem.","section":"Section 4.2, Eq. (11)"},{"comment":"The Taylor coefficients f_p^(k)(t_tau), f_s^(k)(t_tau), and f_q^(k)(t_tau) are never defined as learnable parameters, initialized, or optimized, and the summation limits n, m, and l are never specified. The text says the dominant component is modeled using a third-order Taylor expansion, but the equations are written for arbitrary order and no concrete polynomial fk(t) is ever given. Consequently, the decomposition in Eq. (11) cannot be evaluated or trained as a Taylor polynomial in the sense claimed.","section":"Section 4.3, Eqs. (12)-(14)"},{"comment":"A third-order polynomial plus a single learned residual does not constitute an \"infinite\" Taylor series. There are no higher-order Taylor coefficients, no sequence of terms, and no limiting argument. The time-dependent opacity model in Eq. (7) is likewise not part of the Taylor expansion of the transformation field. The claim that the method forms a complete Taylor series without approximation is therefore not realized by the proposed construction.","section":"Section 4.4, Eqs. (15)-(18) and Section 4.2"},{"comment":"The reported baseline numbers are not credible as printed. D3DGS is reported at 20.53-25.02 PSNR on N3DV in Table 1 but 33.81-38.40 on Technicolor in Table 2, with no explanation of how D3DGS was adapted or configured. Table 1 also reports SCGS at 6.29 PSNR and 0.007 SSIM on Cut Roast Beef, which indicates a broken or misconfigured run. No training protocol, hyperparameters, or evaluation settings are provided for any baseline, undermining the state-of-the-art claim in Sections 5.3 and 5.4.","section":"Section 5.2, Tables 1 and 2"}],"minor_comments":[{"comment":"The Peano remainder is written as Rn(x) = on(x)(x - x0)^n with lim_{x->x0} on(x) = 0; this is an unusual and unclear notation. The standard form is Rn(x) = o((x - x0)^n), or Rn(x) = epsilon(x)(x - x0)^n with epsilon(x) -> 0.","section":"Section 4.1, Eq. (9)"},{"comment":"The coefficients are stated to be in R, but position, scale, and quaternion are vector-valued quantities; the scalar/vector nature of the expansion should be clarified.","section":"Section 4.3, Eqs. (12)-(14)"},{"comment":"The symbol Delta d_j^t is used without definition, and the text alternates between u_i and mu_i for the same LP position; this creates ambiguity in the LBS formula.","section":"Section 4.4, Eq. (17)"},{"comment":"The ablation study reports a single run per configuration with no variance or per-scene breakdown. Given the known sensitivity of 3D Gaussian Splatting training, this limits the strength of the ablation claims.","section":"Section 5.5, Table 3"},{"comment":"The qualitative discussion repeatedly refers to colored boxes (red, blue, green, yellow) that are not clearly visible in the printed figures; please ensure all referenced regions are explicitly marked.","section":"Appendix A.1"},{"comment":"There are several grammatical and typographical errors, including \"the network is struggle to produce\" and \"provides valuable information about the a function model\"; these should be corrected throughout.","section":"Sections 4.1 and 4.2"}],"recommendation":"reject","confidential_remarks":"The empirical pipeline may have practical value, but the advertised mathematical novelty is the load-bearing claim and it is not established. The Peano remainder is not a Peano remainder, the Taylor coefficients are never defined, and the 'infinite' series is not a series. Additionally, the baseline tables appear to contain broken runs, so the quantitative contribution cannot be trusted as currently reported. A resubmission would need to either rigorously derive the Taylor interpretation or abandon that framing and provide a properly documented experimental protocol."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea--tracking Gaussian motion with a third-order polynomial in time on sparse control points, an MLP residual, and LBS blending--is a plausible combination, and the ablation suggests each piece contributes. That part is fine. But the paper wraps it in a mathematical framing that doesn't hold up, and the experiments are presented in a way that makes me trust them less than I'd like.\n\nWhat's genuinely new is narrow: you're taking SC-GS and replacing its deformation network on control points with a time polynomial plus a residual MLP. The paper admits time-conditioned polynomials exist, so the novelty is the specific blend, not the components. That could be a decent incremental recipe if the numbers were believable.\n\nThe problem is that the \"complete Taylor series without approximation\" claim is a tautology. Eq. (11) says any function can be written as a third-order polynomial plus a remainder, and then calls the MLP/LBS output the Peano remainder. Nothing forces that output to vanish faster than (t - t_tau)^k near the expansion center. So it's not a Peano remainder in any useful sense. The coefficients in Eqs. (12)-(14) are never defined as learnable parameters or as derivatives of a concrete function, and the orders n, m, l are left unspecified. Calling this \"infinite\" is also misleading: there's one residual, not a series. The math in Section 4.2-4.4 is not the mathematical foundation the abstract promises.\n\nThen there are the experiments. D3DGS at 20-25 dB on N3DV is roughly 10 dB below its published numbers; 4DGS is also well below. SC-GS gets 6.29 PSNR and 0.007 SSIM on Cut Roast Beef, which is not a model failing, it's either a broken implementation or a mislabeled result. That kind of discrepancy makes the reported gains over \"state of the art\" uninterpretable. No code or training details are given, so I can't check whether the comparison is fair.\n\nThe paper reads like a working incremental method with an overclaimed theoretical wrapper and insufficiently validated comparisons. If the authors stripped out the Taylor mystique, defined the polynomial coefficients concretely, and reran the baselines properly, there might be a solid workshop paper here. As written, the central claim is unsupported and the empirical evidence is shaky. I would not send this to a serious referee in its current form; it needs a major rewrite first.","headline":"The paper's headline claim--a learnable infinite Taylor series with a Peano remainder--does not survive contact with its own equations, and the baseline numbers are far enough off published values to make the empirical results unreliable.","tokens_in":15773,"tokens_out":2193,"would_cite":false,"duration_ms":23788,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a learnable infinite Taylor series for the temporal evolution of 3D Gaussians, splitting motion into a third-order polynomial and a learned remainder, and reports state-of-the-art results on N3DV and Technicolor.","keywords":["dynamic view rendering","3D Gaussian splatting","learnable Taylor series","Peano remainder","deformation field","linear blend skinning","novel view synthesis","temporal modeling"],"falsifier":"Train the model on a dynamic scene and sample the learned remainder $H_k(t)$ and polynomial part $f_k(t)$ along the trajectory of each Gaussian; compute the ratio $H_k(t)/(t-t_\\tau)^k$ for times approaching the expansion center. If the ratio does not tend to zero for a substantial fraction of Gaussians, the object being learned is not a Peano remainder, and the 'complete Taylor series' claim would be refuted even if rendering quality is high.","tokens_in":14535,"feed_emoji":"🎥","tokens_out":8382,"duration_ms":75811,"temperature":0.7,"pith_summary":"The paper tries to establish that the motion of 3D Gaussians in a dynamic scene can be modeled as a learnable infinite Taylor series, one that keeps the interpretability of explicit polynomial trajectories while gaining the flexibility of an end-to-end neural network. Each Gaussian's transformation is split into a third-order Taylor expansion in time, which handles large smooth motion, and a learned residual called the Peano remainder, which captures the rest. On the N3DV and Technicolor light-field datasets the reported PSNR and SSIM numbers are higher than those of prior dynamic Gaussian and NeRF baselines. The sympathetic reading of the central promise is that this decomposition forms a complete Taylor series, so the motion model is not an ad hoc approximation.","feed_headline":"Learnable Taylor series models moving Gaussians for dynamic rendering","feed_subtitle":"Third-order polynomial motion plus a learned residual beats prior dynamic renderers on N3DV and Technicolor","key_machinery":"The load-bearing mechanism is the decomposition in Eq. (11), $T_i(t)=f_k(t)+H_k(t)$, together with the third-order Taylor polynomials for position, scale, and orientation (Eqs. (12)-(14)). The polynomial part is an explicit function of $(t-t_\\tau)$ whose coefficients encode motion at the time center; the Peano remainder part is meant to be the residual that Taylor's theorem leaves after the polynomial, so that the sum behaves like an infinite series. In practice the remainder is generated by a deformation network over a small set of Global Gaussian Primitives and then carried to all other points by Linear Blend Skinning, with distance-based RBF weights that tie nearby points together. This two-part construction is what the paper claims delivers both interpretable large-scale motion and flexible residual correction.","core_discovery":"The central claim is that the transformation of each 4D Gaussian can be written exactly as $T_i(t)=f_k(t)+H_k(t)$, where $f_k(t)$ is a third-order Taylor expansion of position, scale, and rotation around a time center $t_\\tau$, and $H_k(t)$ is the Peano remainder of that expansion. In the paper's construction, the polynomial coefficients are learned, while the remainder is produced by an MLP that predicts offsets for sparse Global Gaussian Primitives, and the offsets are spread to Local Gaussian Primitives through Linear Blend Skinning with Gaussian-kernel RBF weights. The author's claim is that this adds a complete remainder term to the polynomial part, yielding a full Taylor series that estimates the motion model without relying on approximations. The method is evaluated on dynamic novel view rendering, where it reports top quantitative results and crisper reconstructions than comparison systems.","pith_inferences":["Beyond the paper, the mathematical claim can be separated from the rendering result: if the learned $H_k(t)$ does not satisfy the Peano vanishing condition $H_k(t)/(t-t_\\tau)^k\\to 0$ as $t\\to t_\\tau$, the construction is better described as a polynomial plus a flexible neural residual, which may still render well.","Beyond the paper, imposing the vanishing condition explicitly as a regularizer or architectural constraint would convert the Taylor-series framing into a theorem and could make the residual term more predictable.","Beyond the paper, the same decomposition could be applied to other time-varying Gaussian attributes, such as spherical-harmonic coefficients or opacity, since only position, scale, and rotation currently receive Taylor expansions.","Beyond the paper, increasing the polynomial order from three to four and watching whether the learned remainder's contribution shrinks would provide a direct test of whether the expansion is capturing genuinely Taylor-like behavior."],"forward_implications":["Dynamic novel view rendering gains a model that is explicitly interpretable in its main motion terms, so the learned trajectory can be inspected and edited more easily than a pure black-box deformation field.","The explicit third-order polynomial can carry large, smooth motions while the learned remainder fills in non-polynomial detail, which should improve temporal interpolation between captured frames.","Because the remainder is interpolated from sparse control points with rigid link weights, neighboring Gaussians inherit spatial and temporal consistency, reducing flicker and deformation artifacts in rendered video.","The reported results imply the method is a stronger baseline on N3DV and Technicolor than prior dynamic Gaussian splatting and grid-based radiance field approaches."],"supporting_citations":[{"why":"Establishes the static 3D Gaussian splatting representation that this method extends to time-varying scenes.","marker":"[15]"},{"why":"Introduces 4D Gaussian splatting with MLP-predicted deformation fields; serves as the main baseline and the design point the remainder network modifies.","marker":"[39]"},{"why":"Deformable 3D Gaussians, the deformation-field baseline whose implicit approach the paper contrasts with its explicit polynomial part.","marker":"[40]"},{"why":"Sparse-controlled Gaussian splatting, the source of the global/local primitive split and control-point deformation used for the remainder.","marker":"[12]"},{"why":"Provides linear blend skinning, the mechanism that spreads control-point offsets to all local Gaussians in Eq. (17).","marker":"[14]"},{"why":"Supplies the radial basis function formulation used for time-dependent opacity in Eq. (7).","marker":"[3]"},{"why":"Provides the N3DV multi-view video dataset and the DyNeRF baseline used in evaluation.","marker":"[19]"},{"why":"Provides the Technicolor light-field dataset used for the second set of evaluations.","marker":"[32]"}],"fun_headline_variants":["Taylor Gaussians render dynamic scenes with learnable infinite series","Learnable Taylor expansion models Gaussian motion for rendering","Infinite Taylor series for 4D Gaussian dynamics in rendering","Learnable Taylor formula beats prior dynamic renderers","Taylor kernel predicts Gaussian motion for dynamic novel views"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the network output $H_k(t)$, spread by linear blend skinning, actually is the Peano remainder of the Taylor expansion, so that $f_k(t)+H_k(t)$ is a genuine complete series; Taylor's theorem requires the remainder to vanish faster than $(t-t_\\tau)^k$ near the expansion center, and the construction does not enforce that local condition.","fun_headline_variants_meta":{"raw":{"variants":["Taylor Gaussians render dynamic scenes with learnable infinite series","Learnable Taylor expansion models Gaussian motion for rendering","Infinite Taylor series for 4D Gaussian dynamics in rendering","Learnable Taylor formula beats prior dynamic renderers","Taylor kernel predicts Gaussian motion for dynamic novel views"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000174,"raw_usage":{"total_tokens":1276,"prompt_tokens":934,"completion_tokens":342,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":266}},"tokens_in":550,"tokens_out":342,"duration_ms":3466,"temperature":1.0,"reasoning_tokens":266,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:33:36.668627+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the model on a dynamic scene and sample the learned remainder $H_k(t)$ and polynomial part $f_k(t)$ along the trajectory of each Gaussian; compute the ratio $H_k(t)/(t-t_\\tau)^k$ for times approaching the expansion center. If the ratio does not tend to zero for a substantial fraction of Gaussians, the object being learned is not a Peano remainder, and the 'complete Taylor series' claim would be refuted even if rendering quality is high.","supporting_citations":[{"cited_title":"4d gaussian splatting for real-time dynamic scene rendering","cited_arxiv_id":null,"evidence_quote":"Introduces 4D Gaussian splatting with MLP-predicted deformation fields; serves as the main baseline and the design point the remainder network modifies."},{"cited_title":"Deformable 3d gaussians for high- fidelity monocular dynamic scene reconstruction","cited_arxiv_id":null,"evidence_quote":"Deformable 3D Gaussians, the deformation-field baseline whose implicit approach the paper contrasts with its explicit polynomial part."},{"cited_title":"Sc-gs: Sparse-controlled gaussian splatting for editable dynamic scenes","cited_arxiv_id":null,"evidence_quote":"Sparse-controlled Gaussian splatting, the source of the global/local primitive split and control-point deformation used for the remainder."},{"cited_title":"Skinning with dual quaternions","cited_arxiv_id":null,"evidence_quote":"Provides linear blend skinning, the mechanism that spreads control-point offsets to all local Gaussians in Eq. (17)."},{"cited_title":"Neurbf: A neural fields repre- sentation with adaptive radial basis functions","cited_arxiv_id":null,"evidence_quote":"Supplies the radial basis function formulation used for time-dependent opacity in Eq. (7)."},{"cited_title":"Neural 3d video synthesis from multi-view video","cited_arxiv_id":null,"evidence_quote":"Provides the N3DV multi-view video dataset and the DyNeRF baseline used in evaluation."},{"cited_title":"Dataset and pipeline for multi-view light-field video","cited_arxiv_id":null,"evidence_quote":"Provides the Technicolor light-field dataset used for the second set of evaluations."}],"review_version":1}