Rotation-Only Optimization Framework

• Rotation-Only Optimization Framework is a computational strategy that optimizes over rotation matrices on SO(n), ensuring all solutions remain on the manifold.
• It integrates manifold optimization, SDP relaxations, and coordinate descent techniques to achieve robust performance across diverse applications.
• The framework decouples rotations from translations, simplifying complex problems and enhancing scalability in 3D learning, pose estimation, and deep model quantization.

A rotation-only optimization framework refers to any method or algorithm where the variables to be optimized are (or are constrained to be) rotations, typically represented as elements of SO(3) or, more generally, SO(n). Such frameworks are critical whenever the structure of the problem, task, or model is invariant (or equivariant) to translation and scaling, or when other degrees of freedom are decoupled or analytically marginalized. These frameworks have broad impact in computer vision, robotics, wireless communications, 3D learning, and signal processing.

1. Mathematical Foundations and Problem Settings

A rotation-only optimization framework centers the optimization on variables constrained to be rotation matrices—orthogonal matrices with determinant one. Formally, for SO(3), each rotation matrix $R \in \mathbb{R}^{3 \times 3}$ satisfies $R^\top R = I$ and $\det(R) = +1$. Typical scenarios include:

• Rotation averaging: Given a graph of noisy pairwise relative rotation measurements $\{\tilde R_{ij}\}$, the goal is to recover optimal absolute rotations $R_1,\dots, R_n$ that best explain the measurements. The canonical least-squares objective is

$$\min_{R_1,\ldots,R_n \in \mathrm{SO}(3)} \; \sum_{(i,j)\in E} w_{ij} \| R_j R_i^\top - \tilde R_{ij} \|_F^2$$

(Moreira et al., 29 May 2024, Olsson et al., 10 Mar 2025, Moreira et al., 2021, Parra et al., 2021).

• Rotation-only bundle adjustment: Instead of jointly optimizing over rotations, translations, and 3D point positions, the cost is formulated entirely on rotations, e.g., using epipolar or reprojection errors marginalized over translations and/or structure (Lee et al., 2020, Li et al., 16 Nov 2025).
• Rotation-only pose estimation: In decoupled pose frameworks, the rotation is optimized independently or after translation is estimated (Yang et al., 2022).
• Deep network quantization and transformation: Here, a rotation-only framework is used to reorient weight matrices or channel axes prior to quantization or for improved robustness (Choi et al., 2 May 2025).
• Channel optimization for movable antennas: The rotation-only step optimizes device orientation to maximize link quality, with positions held fixed during this phase (Jiang et al., 29 Apr 2025).

The key mathematical structure in all these settings is the non-convex geometry of the rotation group SO(n), which leads to specialized optimization schemes, relaxations, or closed-form solutions tailored to the rotation manifold.

2. Core Methodologies and Algorithmic Approaches

Rotation-only frameworks admit a range of methodologies:

a. Manifold Optimization

Manifold-based solvers exploit the fact that the set of rotations forms a compact Lie group. Typical strategies involve:

• Parameterization: Axis-angle vectors, quaternions, or direct matrix representations.
• Updates: Optimization proceeds in the tangent space (Lie algebra), with updates mapped back via the exponential map:

$R \leftarrow \exp([\delta\phi]_\times) R$

This ensures all iterates remain on SO(3) (Lee et al., 2020, Yang et al., 2022, Li et al., 16 Nov 2025).

• Gradient and Hessian computation: For least-squares formulations, Gauss–Newton or Levenberg–Marquardt methods are standard.

b. Semidefinite Programming (SDP) Relaxations

SDP relaxations are foundational for certifiable global solutions:

• Lifted variable approach: Represent $R$ via $Y = R R^\top$, leading to a convex feasible set with block-diagonal constraints $Y_{ii} = I_n$ (Moreira et al., 29 May 2024, Olsson et al., 10 Mar 2025, Moreira et al., 2021).
• Convex hull constraints: For anisotropic objectives, further constraints (e.g., blockwise convex hull of SO(3)) rule out spurious reflections and ensure tightness (Olsson et al., 10 Mar 2025).
• Certification: The SDP solution is globally optimal if the optimal $Y^*$ is rank $n$ (or $3$ for SO(3)).

c. Coordinate Descent and Primal–Dual Methods

• Coordinate Descent: Update one rotation at a time by blockwise or coordinate sampling, directly minimizing the local subproblem (often via closed-form SVDs or analytical minimization over SO(3)) (Parra et al., 2021, Shalit et al., 2013).
• Primal–Dual: Alternates between minimizing a Lagrangian over the manifold and updating the dual (multiplier) variables, with convergence and optimality monitored via KKT conditions. Closed-form solutions arise in certain graph topologies (e.g., cycles) (Moreira et al., 2021, Moreira et al., 29 May 2024).

d. Specialized Methods for Applications

• Grouped Sequency-arranged Rotations (GSR): Block-diagonal rotations composed of grouped, sequency-ordered Walsh–Hadamard blocks minimize quantization error in deep network PTQ without training (Choi et al., 2 May 2025).
• Adversarial rotation optimization: In robust 3D learning, adversarial rotations maximize classifier loss, found via gradient-based updates in axis-angle or Euler angles, with group structure exploited for efficiency (Wang et al., 2022).
• Hidden convexity and convex reformulation: Exploits the convexity of certain linear images or projections of SO(n) to reformulate single-constraint or low-rank constrained problems exactly as convex programs (Ramachandran et al., 2023).

3. Representative Frameworks and Their Properties

The following table summarizes major frameworks, their core techniques, and application domains:

Framework / Paper	Core Technique	Domain / Application
Primal–dual rotation averaging (Moreira et al., 29 May 2024, Moreira et al., 2021)	Spectral + dual update, certificate	SfM, SLAM, pose-graph optimization
Certifiable anisotropic rotation averaging (Olsson et al., 10 Mar 2025)	SDP + blockwise convex-hull constraint	SfM under anisotropic uncertainty
Rotation coordinate descent (Parra et al., 2021)	Sparse coordinate updates, SVD	Large-scale rotation averaging
Bundle Adjustment (ROBA) (Lee et al., 2020)	Adam on SO(3), epipolar cost	Multiview geometry, structure-from-motion
Givens coord-descent (Shalit et al., 2013)	Givens/sparse rotation, 1-D subproblems	Orthogonal matrix optimization, PCA/tensor
GSR for quantization (Choi et al., 2 May 2025)	Walsh–Hadamard, block-diagonal grouping	LLM/PTQ, inference acceleration
Adversarial rotation (ART-Point) (Wang et al., 2022)	Axis-wise PGD, class-wise assignment	3D point cloud robustness
Hidden convexity (Ramachandran et al., 2023)	2D convex projection, ellipsoid/SDP	Wahba’s prob., pose w/extra constraints
6DMA antenna control (Jiang et al., 29 Apr 2025)	Euler angle gradient ascent, projected	Wireless comms, channel optimization

Each embodies the rotation-only principle via stringent optimization on the SO(n) group, thereby leveraging structure for algorithmic efficiency, statistical robustness, and sometimes closed-form recovery.

4. Special Topics: Decoupling, Certifiability, and Closed-Forms

• Decoupling translation and structure: Key advances in SfM and pose estimation demonstrate how, with sufficient correspondences, translation can be expressed analytically in terms of rotation. This enables rotation-only frameworks that are immune to errors in translation or structure (Li et al., 16 Nov 2025, Lee et al., 2020).
• Certifiability and tightness: Strong duality/tightness results guarantee that solutions of SDP relaxations coincide with global minimizers of the original nonconvex problem in a broad parameter regime (especially under bounded noise or for certain graph topologies such as cycles) (Moreira et al., 29 May 2024, Moreira et al., 2021, Olsson et al., 10 Mar 2025).
• Closed-form solutions: In cycle-structured measurement graphs, the stationary points and minimizers of the rotation averaging objective can be characterized analytically, e.g., via functions of the cycle error and its $n$th roots (Moreira et al., 2021, Moreira et al., 29 May 2024).

5. Applications and Practical Impact

• Vision and Robotics: Pose-graph optimization, multiview geometry, SfM, and SLAM pipelines commonly integrate rotation-only stages for global initialization or refinement. Evidence shows these steps deliver order-of-magnitude speed and accuracy improvements (Moreira et al., 29 May 2024, Parra et al., 2021).
• LLM Quantization: Deployments of large models at ultra-low precision benefit from rotation-only transformations to minimize quantization loss, as in the GSR method, which is strictly training-free yet competitive with optimized learned rotations (Choi et al., 2 May 2025).
• Communications and Sensing: Channel optimization in 6DMA considers antenna rotations to maximize statistical rate, solved efficiently in a rotation-only subroutine (Jiang et al., 29 Apr 2025).
• Pose Estimation and Perception: In multi-view 6D object pose estimation or odometry, rotation-only optimization provides more robust, more accurate, and faster inference, especially when symmetries and multimodality are present (Yang et al., 2022, Muhle et al., 2022).

6. Computational Complexity, Limitations, and Extensions

Rotation-only frameworks often exploit problem structure for computational gains:

• Sparse updates and memory: Methods such as coordinate-descent avoid forming large dense SDP matrices, achieving $O(m)$ per-iteration cost for $m$ measurements (Parra et al., 2021).
• Scalability: First-order conic methods and primal–dual updates scale to thousands of cameras or devices in seconds (Moreira et al., 29 May 2024).
• Closed-form subproblems: Givens rotation coordinate-descent yields $O(d)$ updates on $d\times d$ matrices, vital for large $d$ (Shalit et al., 2013).
• Limitations: Most frameworks assume well-conditioned measurement graphs or sufficient non-degeneracy (e.g., not all views coplanar), and some only certify optimality under mild noise (Moreira et al., 2021).
• Extensions: Hidden convexity approaches expand the range of exactly-solvable rotation problems with additional linear constraints, as long as they respect the underlying convexity conditions (Ramachandran et al., 2023).
• Applicability to higher SO(n): While most practical work focuses on SO(3), several results generalize to arbitrary SO(n).

7. Empirical Validation and Summary of Gains

Rotation-only methods have shown:

• Statistically significant improvement in rotation estimation accuracy—median error reductions of up to 80% over initialization-only or translation-coupled methods (Lee et al., 2020, Li et al., 16 Nov 2025).
• Speedups of 10–100× over previous certifiable solvers in both synthetic and real datasets, especially for coordinate-descent and primal–dual methods (Parra et al., 2021, Moreira et al., 29 May 2024).
• Plug-and-play deployment in quantitative frameworks—the GSR rotation is applied without retraining and yields 2× improvement in perplexity in 2-bit quantization for Llama-2-7B (Choi et al., 2 May 2025).
• Robustness and accuracy under real-time constraints—rotation-only pipelines in visual odometry achieve 19% drift reduction in KITTI at real-time speeds (Muhle et al., 2022).

Empirical data continually affirms that, for a wide class of geometric, learning, and statistical problems, rotation-only optimization frameworks enable precise, fast, and certifiable solutions that efficiently harness group structure and problem decoupling.