Learnable Infinite Taylor Gaussian for Dynamic View Rendering

Bingbing Hu; Bo Xu; Gim Hee Lee; Haoye Dong; Junfeng Yao; Rui Xie; Yanyan Li

REVIEW 4 major objections 6 minor 47 references

Learnable Infinite Taylor Gaussian for Dynamic View Rendering

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.04282 v2 pith:ZVHR623K submitted 2024-12-05 cs.CV

Bingbing Hu , Yanyan Li , Rui Xie , Bo Xu , Haoye Dong , Junfeng Yao , Gim Hee Lee This is my paper · ORCID

classification cs.CV

keywords dynamicviewrendering3DGaussiansplattinglearnableTaylorseriesPeanoremainderdeformationfieldlinearblendskinningnovelsynthesistemporalmodeling

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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 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.

What'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.

The 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.

Then 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.

The 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.

Referee Report

4 major / 6 minor

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.

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 (4)

[Section 4.2, Eq. (11)] 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 4.3, Eqs. (12)-(14)] 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 4.4, Eqs. (15)-(18) and Section 4.2] 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 5.2, Tables 1 and 2] 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.

minor comments (6)

[Section 4.1, Eq. (9)] 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 4.3, Eqs. (12)-(14)] 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 4.4, Eq. (17)] 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 5.5, Table 3] 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.
[Appendix A.1] 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.
[Sections 4.1 and 4.2] 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.

Circularity Check

2 steps flagged · score 6.0 of 10

The 'complete Taylor series' claim is a tautology: the Peano remainder H_k is defined as whatever the MLP/LBS network outputs, so Eq. (11) is an identity plus a fitted residual rather than a Taylor expansion.

self definitional [Section 4.2, Eq. (11); Section 4.4, Eqs. (15)-(18)]
"The Peano remainder is constructed via the deformation field, forming a complete Taylor series to estimate the motion model without approximation. ... Ti(t) = fk(t) + Hk(t) (11) ... The Peano Remainder for the LP points is interpolated using Linear Blend Skinning (LBS) [14]."

Taylor's theorem requires the Peano remainder to satisfy Hk(t) = o((t - t_tau)^k) as t -> t_tau, but Eqs. (15)-(18) impose no such locality or vanishing condition: Hk is an MLP decoder output blended by LBS. Thus Eq. (11) is the trivial identity T_i = f_k + H_k for any fitted H_k, and calling H_k the 'Peano remainder' does not make the decomposition a Taylor series. The 'complete series without approximation' claim is true by construction, not by Taylor's theorem.
renaming known result [Section 4.4, Eqs. (17)-(18)]
"The Peano remainder terms of the motion equation for LP points can be accurately estimated using LBS via the following function: Delta_mu_i^t = sum_j w_ij ( R_j^t (mu_i - p_j) + p_j + Delta_d_j^t ) (17)"

This is the standard sparse-control plus linear blend skinning deformation recipe used in prior dynamic Gaussian work (e.g., SC-GS), not a remainder derived from Taylor's theorem. Renaming the network residual a 'Peano remainder' and the finite polynomial-plus-residual recipe a 'learnable infinite Taylor Formula' does not constrain the model or yield the advertised infinite Taylor series; it repackages a known empirical construction under Taylor-series vocabulary.

full rationale

The empirical evaluation against external N3DV/Technicolor benchmarks is self-contained; no fitted parameter is relabeled as a prediction, and self-citations [11, 20, 44, 45] are contextual, not load-bearing. The circularity is confined to the theoretical framing: the central advertised result, a 'complete/infinite Taylor series,' is obtained by defining the residual (MLP plus LBS output) as the Peano remainder and writing T_i = f_k + H_k, which is an algebraic identity true for any f_k and H_k. No property of Taylor's theorem (vanishing remainder, higher-order coefficients, convergence) is used or checked, and Eqs. (12)-(14) never specify the derivative coefficients or summation orders. Hence the derivation chain reduces to a label; score 6 rather than 8 because the rendering system itself has independent empirical content and does not depend on the Taylor claim for its reported numbers.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central 'Taylor' contribution depends on unspecified per-Gaussian Taylor coefficients, a mislabeled Peano remainder network, and domain assumptions inherited from SC-GS. There are no new physical entities, but the 'Peano remainder' is an invented entity with no independent evidence.

free parameters (5)

Taylor coefficients f_p^(k), f_s^(k), f_q^(k) per Gaussian = not stated
These appear in Eqs. 12-14 but the paper does not specify how they are initialized, optimized, or constrained. They are per-Gaussian parameters fitted to the training views, and the central 'Taylor' claim depends on them.
MLP weights for GP offsets = learned
Eq. 15 defines Delta_GP = MLP(GP); the network weights determine the deformation of global primitives and are fitted with backpropagation.
RBF kernel radius r_j = learned
In Eq. 16, the radius r_j controls the LBS weight decay; the paper says it is learned by gradient descent.
Opacity model parameters sigma^s_i, s^tau_i, mu^tau_i = learned
Eq. 7 models temporal opacity with three per-Gaussian parameters; they are fitted and affect visibility during rendering.
Taylor expansion center t_tau = not specified
The time center around which the expansion is made must be selected or learned, but the paper does not describe how t_tau is chosen.

assumptions (4)

ad hoc to paper Taylor's theorem with Peano remainder is applicable to Gaussian trajectory functions.
Section 4.1 states the theorem, but Section 4.4 replaces the remainder with an MLP/LBS output that does not satisfy Peano's condition; the theorem is not actually used.
domain assumption Global and local Gaussian primitives maintain stable correspondence across views and time.
Section 4.4 assumes GPs remain stable so their offsets can drive LP deformation through LBS; this is imported from SC-GS and is not verified.
domain assumption Temporal opacity follows a radial basis function (Eq. 7).
The RBF opacity model is an ad hoc modeling choice, fitted to data, and is not derived from the rendering equations.
domain assumption Quaternions can be expanded as a plain polynomial Taylor series (Eq. 14).
Quaternions used for rotation must stay unit-norm; a raw polynomial expansion does not preserve this constraint, and the paper does not specify a normalization or re-projection step.

invented entities (1)

Peano remainder network
purpose: To absorb the gap between the third-order polynomial and the true motion, letting the paper claim a complete Taylor series.
The network is trained on the same data, provides no out-of-sample prediction, and does not obey the local smallness property of Peano's remainder, so it is a mislabeled fitted module rather than a derived entity.

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Cite this review

Pith. "Pith review of Learnable Infinite Taylor Gaussian for Dynamic View Rendering." pith.science (2026). https://pith.science/paper/ZVHR623K

@misc{pith2026241204282,
  author       = {Pith},
  title        = {Pith review of: Learnable Infinite Taylor Gaussian for Dynamic View Rendering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVHR623K}},
  note         = {Machine review of arXiv:2412.04282}
}

read the original abstract

Capturing the temporal evolution of Gaussian properties such as position, rotation, and scale is a challenging task due to the vast number of time-varying parameters and the limited photometric data available, which generally results in convergence issues, making it difficult to find an optimal solution. While feeding all inputs into an end-to-end neural network can effectively model complex temporal dynamics, this approach lacks explicit supervision and struggles to generate high-quality transformation fields. On the other hand, using time-conditioned polynomial functions to model Gaussian trajectories and orientations provides a more explicit and interpretable solution, but requires significant handcrafted effort and lacks generalizability across diverse scenes. To overcome these limitations, this paper introduces a novel approach based on a learnable infinite Taylor Formula to model the temporal evolution of Gaussians. This method offers both the flexibility of an implicit network-based approach and the interpretability of explicit polynomial functions, allowing for more robust and generalizable modeling of Gaussian dynamics across various dynamic scenes. Extensive experiments on dynamic novel view rendering tasks are conducted on public datasets, demonstrating that the proposed method achieves state-of-the-art performance in this domain. More information is available on our project page(https://ellisonking.github.io/TaylorGaussian).

Figures

Figures reproduced from arXiv: 2412.04282 by the authors.

**Figure 1.** The detailed architecture of the proposed method. The framework includes Gaussian Initialization, Sparse Point Sampling, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Comparison of novel view rendering on the N3DV dataset, with problem regions highlighted in boxes. More results can be found [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Qualitative analysis of novel view rendering on the N3DV dataset, comparing the detail information of reconstructed images [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Sear Steak Novel View Rendering on the N3DV Dataset: Qualitative Analysis of Ablation Experiments - Comparison of Reconstruction Quality and Detail Representation with Module Ablations. B. Performance Analysis of Large-Scale Data Scene Reconstruction To better evaluate…

**Figure 5.** Figure 5: Qualitative analysis of novel view rendering on the Birthday dataset from the Technicolor, comparing the detailed reconstructions [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Qualitative analysis of novel view rendering on the Painter dataset from the Technicolor, comparing the detailed reconstructions [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

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