Pith. sign in

REVIEW 2 major objections 4 minor 15 references

When joint rotations have known bounds, Euler angles plus Kolmogorov-Arnold Networks regress orientation more accurately and efficiently than the usual 6D-plus-MLP pipeline.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-13 01:27 UTC pith:KIECASIP

load-bearing objection Solid, practical result: under known joint limits, KAN+Euler beats matched MLP+6D on accuracy and efficiency, with theory and ablations that line up. the 2 major comments →

arxiv 2607.09650 v1 pith:KIECASIP submitted 2026-07-10 cs.CV

Revisiting Euler-Angle Regression with Kolmogorov-Arnold Networks

Yangting Sun , Zijun Cui , Yufei Zhang This is my paper
classification cs.CV
keywords Euler anglesrotation regressionKolmogorov-Arnold Networksinverse kinematics6D rotation representationbounded joint rangesobject pose estimationbiomechanical models
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Many robots and biomechanical models already parameterize rotations as Euler angles with tight, known range limits. Standard practice still avoids regressing those angles directly, preferring continuous overparameterized forms such as the 6D representation decoded by ordinary multilayer perceptrons. This paper argues that the real obstacle is not Euler angles themselves but the mismatch between fixed-activation networks and the angular structure of bounded Euler targets. By pairing range-aware Euler modeling (intervals that exclude 2π wrapping and a middle-axis ordering that keeps the gimbal-prone angle inside (−π/2, π/2)) with Kolmogorov-Arnold Networks, whose edges carry learnable univariate splines, the regression target becomes near-additive and is therefore a natural fit for KAN. Controlled rotation experiments, object pose estimation, robotic-arm inverse kinematics, and hand and full-body biomechanical inverse kinematics all show lower angle and geodesic error, faster convergence, and better parameter efficiency than the matched MLP+6D baseline. The result matters because it restores the native, interpretable coordinates of articulated systems without sacrificing training stability.

Core claim

In the practical regime where each Euler angle is confined to a known compact interval free of periodic wrapping and singularities, the map from observations to joint angles decomposes into a dominant additive part plus a small interaction residual. Kolmogorov-Arnold Networks, which place learnable univariate spline functions on edges rather than fixed activations on nodes, approximate that additive structure more accurately and with fewer parameters than multilayer perceptrons operating on 6D or other continuous embeddings. The pairing of range-aware Euler charts with KAN therefore yields consistently lower mean angle and geodesic error, faster optimization, and higher sample efficiency acr

What carries the argument

Range-aware Euler chart plus Kolmogorov-Arnold Network: admissible intervals exclude 2π ambiguity, the most-constrained axis is placed in the middle Euler slot so it stays inside (−π/2, π/2), and KAN’s additive spline edge functions then realize the near-additive ANOVA structure that the bounded chart induces, with an explicit approximation bound of the form d·C·M·K^(−p)+ϵ_int.

Load-bearing premise

The method assumes that every joint’s admissible angle intervals are known in advance, fit inside one 2π period, and can be ordered so the most-restricted axis sits in the middle Euler slot and never reaches ±90 degrees.

What would settle it

Re-run the controlled range-sweep and the hand or arm inverse-kinematics benchmarks with the same range constraints and axis ordering: if KAN+Euler fails to undercut matched-capacity MLP+6D on mean angle and geodesic error once the middle axis is kept strictly inside (−π/2, π/2), or if the measured interaction residual does not shrink as the ranges tighten, the central claim is false.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper revisits Euler-angle regression for 3D rotations in the practical regime of articulated systems (robots, biomechanical models) where joint angles have known, bounded ranges. It argues that effectiveness depends on the interaction of representation, architecture, and domain constraints, and proposes combining range-aware Euler modeling (with constraint-aware axis ordering to keep the middle angle inside (−π/2, π/2)) with Kolmogorov–Arnold Networks (KAN), whose learnable univariate edge functions match a near-additive target structure. Lemma 1 and Theorem 1 formalize this via ANOVA decomposition plus a BCH expansion showing that higher-order interaction residual ϵ_int is O(δ²) in the chart half-width, yielding a spline approximation bound that favors KAN over generic MLP guarantees. Experiments on controlled matrix-to-Euler regression, ModelNet chair pose estimation, Franka Panda IK (3–5 DoF), FreiHand MANO IK, and SKEL full-body articulation report consistent gains in MAE/GE/FKE, convergence, and parameter efficiency over matched-capacity MLP+6D (and AA) baselines, with ablations on range (div), constraints, and data fraction.

Significance. If the result holds, the work supplies a concrete, architecture-aware alternative to the dominant MLP+6D pipeline for the large class of systems that already operate in bounded joint Euler coordinates. The theoretical link between range-constrained Euler charts and KAN’s additive spline form is a useful inductive-bias argument, and the multi-domain empirical package (controlled, object, robotic, hand, body) with matched parameters, multi-seed reporting, and range/constraint ablations makes the claim falsifiable and practically actionable. Public code is promised. The contribution is incremental rather than foundational—KAN itself is prior work, and the regime is deliberately restricted—but it is a clear, well-scoped advance for rotation regression in articulated systems.

major comments (2)
  1. The central claim is scoped to a priori known, singularity-free ranges with middle-axis ordering (Sec. 3.4, App. A). When this fails (div=1 in Figs. 2b and 4; shoulder axial rotation reaching ±π/2 in App. A), performance collapses as predicted. The manuscript should state more explicitly in the abstract/introduction and conclusion that the method is not a general SO(3) solution and should quantify how often real systems satisfy the middle-axis condition (or provide a fallback when they do not).
  2. Lemma 1 / Theorem 1: the BCH argument motivates that ϵ_int shrinks with δ, but the bound is not derived from first principles for the regression map f:X→Θ; it is an existence-style ANOVA+spline bound. A short empirical measurement of ϵ_int (or of mixed partials) versus div in App. C would strengthen the theory–experiment link that the paper already claims to confirm.
minor comments (4)
  1. Table 5(a) (controlled regression): KAN+AA is numerically best on matrix input; the text correctly notes this is task-specific, but a one-sentence clarification in the main text (not only the appendix) would prevent misreading of the primary claim.
  2. Inference latency is reported only in App. C (905 µs vs 36 µs); a brief note in the efficiency discussion of Sec. 4.1 would make the FLOPs-vs-wall-clock trade-off transparent.
  3. Notation: θ, α/β/γ, and (θ1,θ2,θ3) are used interchangeably for the same triple; a single consistent notation after Eq. (2) would improve readability.
  4. Related work could briefly situate recent SO(3) representation analyses (e.g., Geist et al., Schuck et al.) against the architecture–constraint interaction emphasized here.

Circularity Check

0 steps flagged

No significant circularity: approximation bounds rest on classical spline/ANOVA/BCH arguments; empirical gains are held-out comparisons, not fits renamed as predictions.

full rationale

The paper’s load-bearing chain is: (i) practical joint limits plus middle-axis ordering yield a compact, singularity-free Euler chart Θ (Sec. 3.4); (ii) on that chart the Euler-to-rotation map is near-additive by BCH, so the regression target has small interaction residual ϵ_int (Lemma 1); (iii) KAN’s additive spline subclass then admits the bound d Cp M K^{-p} + ϵ_int (Theorem 1), which is more parameter-efficient than generic MLP rates when ϵ_int is small (Prop. 1). None of these steps define the claimed quantity in terms of itself, fit a free parameter to the reported metric, or import a uniqueness theorem from the authors. The bound is classical approximation theory (de Boor; ANOVA–Sobolev) plus a standard Lie-group expansion; it does not embed experimental outcomes. Empirical sections train networks and evaluate MAE/GE/FKE on held-out data against external baselines (MLP+6D, AA, etc.), including controlled range sweeps that falsify the method when ranges are full (div=1) and strengthen it as ranges tighten—behavior predicted by the theory rather than forced by construction. Self-citations (Zhang et al. range tables for hand/body) supply domain intervals used as scope conditions, not a circular proof of accuracy. Score 1 only for that minor, non-load-bearing self-citation of prior range tables; the central derivation and claims are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central claim rests on classical approximation theory, the Kolmogorov–Arnold representation idea, known SO(3) geometry, and published biomechanical/robot joint limits. No new physical entities are postulated; free parameters are ordinary network hyper-parameters and literature-derived ranges.

free parameters (3)
  • spline degree p and number of basis functions K_ℓ
    Chosen by the authors (typically p=3, K≈4–12) to control edge expressivity; not derived from first principles.
  • division factor div that scales Euler ranges
    Hand-chosen experimental knob used to sweep range tightness; the theory only requires that ranges be bounded.
  • network widths and learning-rate schedules
    Matched across models for fairness but still free design choices that affect absolute numbers.
axioms (6)
  • standard math Kolmogorov–Arnold representation theorem (existence of univariate edge functions that can represent continuous multivariate maps)
    Invoked to motivate the KAN architecture (Section 3.3).
  • standard math Classical degree-p spline approximation rate O(K^{-p}) on compact intervals
    Used in the proof of Theorem 1 (Appendix B, citing de Boor).
  • standard math Functional ANOVA decomposition and Sobolev-type control of higher-order interactions by mixed second derivatives
    Lemma 1 (Appendix B, citing Rabitz & Aliş, Hooker).
  • standard math Baker–Campbell–Hausdorff expansion of products of elemental rotations yields additive first-order terms and O(δ²) commutators
    Used to motivate near-additivity under small chart half-width δ (Section 3.5).
  • domain assumption Published anatomical/mechanical joint limits for MANO hand, SKEL body, and Franka Panda are accurate and known a priori
    Appendix A and Section 3.4; the singularity-free chart construction depends on them.
  • domain assumption The most-constrained axis of each joint can be placed in the middle Euler slot while remaining inside (−π/2, π/2)
    Section 3.4; shoulder is noted as a boundary case.

pith-pipeline@v1.1.0-grok45 · 23554 in / 2679 out tokens · 32177 ms · 2026-07-13T01:27:04.657054+00:00 · methodology

0 comments
read the original abstract

In many real-world systems, including articulated robots and biomechanical models, rotations are defined in joint space and naturally parameterized by Euler angles with bounded ranges. Yet regressing Euler angles remains challenging, as their discontinuities and singularities often destabilize training. In this work, we revisit Euler-angle regression and show that its effectiveness depends critically on the interaction between rotation representation, regression architecture, and domain constraints. We introduce a new framework that combines range-aware Euler modeling with Kolmogorov-Arnold Networks (KAN), which replace fixed node-wise activations with learnable univariate functions on edges. We further provide theoretical analysis indicating that bounded Euler ranges motivate a near-additive structure in the regression function, which favors the additive functional form of KAN, and we confirm this trend empirically. Extensive experiments on controlled rotation regression, object pose estimation, and robotic and human inverse kinematics demonstrate consistent improvements in accuracy, convergence, and efficiency. The code will be publicly available.

Figures

Figures reproduced from arXiv: 2607.09650 by Yangting Sun, Yufei Zhang, Zijun Cui.

Figure 1
Figure 1. Figure 1: Illustration of Kolmogorov-Arnold Networks (KAN) and their advantages over MLPs. We show a two-input, one-hidden-layer example (green nodes), although practical models are typically deeper and wider. Euler angles are highly interpretable, as each parameter represents a rotation about a fixed axis and the rotation matrix has an explicit trigonometric structure in (α, β, γ). However, singularities occur at β… view at source ↗
Figure 2
Figure 2. Figure 2: Object pose estimation. (a) GE vs. training iteration for KAN+Euler, MLP+6D, and MLP+Euler at div=1.3. (b) GE vs. div. Axes are log-scaled. performs poorly because sampled rotations reach the ±π/2 gimbal set, where the Euler chart itself is ill￾behaved; once the range excludes this singularity, KAN+Euler crosses below MLP+6D near div=1.1, and the margin widens to 1.6× by div=2 as tighter ranges make the ta… view at source ↗
Figure 4
Figure 4. Figure 4: Effect of Euler-angle range. GE vs. div; axes are log-scaled. When the full range is used (div = 1), both Euler variants perform poorly: KAN+Euler (3.99 GE) and MLP+Euler (6.01) trail MLP+6D (0.28), because sam￾ples near the range boundaries still suffer from discontinu￾ities and singularities. As the range tightens, KAN+Euler improves sharply, crossing below MLP+6D near div=1.3 and reaching 3.9× lower err… view at source ↗
Figure 3
Figure 3. Figure 3: Optimization on controlled rotation regression. Both models are trained with the same angle-space MSE loss. Left: training loss (log scale); right: Euler-angle error (MAE, degrees) [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 8 linked inside Pith

  1. [1]

    Object-centric dexterous manipulation from human motion data.arXiv preprint arXiv:2411.04005,

    Yuanpei Chen, Chen Wang, Yaodong Yang, and C Karen Liu. Object-centric dexterous manipulation from human motion data.arXiv preprint arXiv:2411.04005,

    Pith/arXiv arXiv

  2. [2]

    Opensim: open-source software to create and analyze dynamic simulations of movement.IEEE transactions on biomedical engineering, 54(11):1940–1950,

    Scott L Delp, Frank C Anderson, Allison S Arnold, Peter Loan, Ayman Habib, Chand T John, Eran Guendelman, and Darryl G Thelen. Opensim: open-source software to create and analyze dynamic simulations of movement.IEEE transactions on biomedical engineering, 54(11):1940–1950,

    1940

  3. [3]

    Learning with 3d rotations, a hitchhiker’s guide to so (3).arXiv preprint arXiv:2404.11735,

    A René Geist, Jonas Frey, Mikel Zhobro, Anna Levina, and Georg Martius. Learning with 3d rotations, a hitchhiker’s guide to so (3).arXiv preprint arXiv:2404.11735,

    Pith/arXiv arXiv

  4. [4]

    Learning unorthogonalized matrices for rotation estimation.arXiv preprint arXiv:2312.00462,

    Kerui Gu, Zhihao Li, Shiyong Liu, Jianzhuang Liu, Songcen Xu, Youliang Yan, Michael Bi Mi, Kenji Kawaguchi, and Angela Yao. Learning unorthogonalized matrices for rotation estimation.arXiv preprint arXiv:2312.00462,

    Pith/arXiv arXiv

  5. [5]

    Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Soljačić, Thomas Y Hou, and Max Tegmark

    URLhttps://openreview.net/ forum?id=fVCRBAXJ21. Ziming Liu, Yixuan Wang, Sachin Vaidya, Fabian Ruehle, James Halverson, Marin Soljačić, Thomas Y Hou, and Max Tegmark. KAN: Kolmogorov-Arnold networks.arXiv preprint arXiv:2404.19756,

    Pith/arXiv arXiv

  6. [6]

    From kepler to newton: Inductive biases guide learned world models in transformers.arXiv preprint arXiv:2602.06923,

    Ziming Liu, Sophia Sanborn, Surya Ganguli, and Andreas Tolias. From kepler to newton: Inductive biases guide learned world models in transformers.arXiv preprint arXiv:2602.06923,

    arXiv

  7. [7]

    On the role of rotation equivariance in monocular 3d human pose estimation.arXiv preprint arXiv:2601.13913,

    Pavlo Melnyk, Cuong Le, Urs Waldmann, Per-Erik Forssén, and Bastian Wandt. On the role of rotation equivariance in monocular 3d human pose estimation.arXiv preprint arXiv:2601.13913,

    Pith/arXiv arXiv

  8. [8]

    A smooth representation of belief over so (3) for deep rotation learning with uncertainty.arXiv preprint arXiv:2006.01031,

    Valentin Peretroukhin, Matthew Giamou, David M Rosen, W Nicholas Greene, Nicholas Roy, and Jonathan Kelly. A smooth representation of belief over so (3) for deep rotation learning with uncertainty.arXiv preprint arXiv:2006.01031,

    Pith/arXiv arXiv 2006

  9. [9]

    A primer on so (3) action representations in deep reinforcement learning.arXiv preprint arXiv:2510.11103,

    Martin Schuck, Sherif Samy, and Angela P Schoellig. A primer on so (3) action representations in deep reinforcement learning.arXiv preprint arXiv:2510.11103,

    Pith/arXiv arXiv

  10. [10]

    Are euler angles a useful rotation parameterisation for pose estimation with normalizing flows?arXiv preprint arXiv:2511.02277,

    Giorgos Sfikas, Konstantina Nikolaidou, Foteini Papadopoulou, George Retsinas, and Anastasios L Kesidis. Are euler angles a useful rotation parameterisation for pose estimation with normalizing flows?arXiv preprint arXiv:2511.02277,

    arXiv

  11. [11]

    Cameras as rays: Pose estimation via ray diffusion.arXiv preprint arXiv:2402.14817, 2024a

    Jason Y Zhang, Amy Lin, Moneish Kumar, Tzu-Hsuan Yang, Deva Ramanan, and Shubham Tulsiani. Cameras as rays: Pose estimation via ray diffusion.arXiv preprint arXiv:2402.14817, 2024a. Yufei Zhang, Hanjing Wang, Jeffrey O Kephart, and Qiang Ji. Body knowledge and uncertainty modeling for monocular 3d human body reconstruction. InProceedings of the IEEE/CVF I...

    Pith/arXiv arXiv

  12. [12]

    •Appendix B: Theoretical Analysis of Euler-Angle Regression with KAN

    14 Revisiting Euler-Angle Regression with Kolmogorov-Arnold Networks Supplementary Material In this supplementary material, we first provide additional analysis and specifications underlying our method: •Appendix A: Per-Joint Range Specifications and Singularity Handling. •Appendix B: Theoretical Analysis of Euler-Angle Regression with KAN. We then report...

    2017

  13. [13]

    Rep.” refers to the employed rotation repre- sentation, and “Params

    C Controlled Rotation Regression Setup.Following Zhou et al. (2019), we study rotation regression in a controlled setting while isolating complexity from downstream tasks. Given a3×3rotation matrixR∈SO(3)as input, we train neural networks to recover the corresponding ZXY Euler angles. Following the axis-assignment guideline of Sec- tion 3.4, the gimbal-pr...

    2019

  14. [14]

    Figure 4:Effect of Euler-angle range.GE vs.div; axes are log-scaled. When the full range is used (div = 1), both Euler variants perform poorly: KAN+Euler (3.99 GE) and MLP+Euler (6.01) trail MLP+6D (0.28), because sam- ples near the range boundaries still suffer from discontinu- itiesandsingularities. Astherangetightens, KAN+Euler improves sharply, crossi...

    2023

  15. [15]

    The output is the vector of active joint angles, and all models are trained with MSE loss in joint-angle space

    or the first two columns of the rotation matrix for 6D (input dimension 9). The output is the vector of active joint angles, and all models are trained with MSE loss in joint-angle space. All models are matched to approximately 100K parameters and trained for 30 epochs with batch size 1,024, using AdamW with learning rate10−3. Evaluation is performed ever...

    2017