发表机构
University of California, Riverside（加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对SO(3)上的梯度流，构造了一个经严格认证的反例，证明测地强凸性不蕴含测地球的前向不变性，揭示了强凸性约束梯度投影方向的机制。
AI 中文摘要
设$\boldsymbol{\text{mathcal{C}}=\boldsymbol{\text{overline{\text{mathcal{B}}}}}_{\rho}(R_c)$为双不变度量下SO(3)中半径$\rho<\boldsymbol{\text{π}}/2$的测地球，$f$在$\boldsymbol{\text{mathcal{C}}}$上是具有内部极小值点的测地强凸函数。人们很容易期望梯度流$\boldsymbol{\text{dot R}}=R(-\nabla f)^\boldsymbol{\text{wedge}}$能保持$\boldsymbol{\text{mathcal{C}}}$的前向不变性：该流被吸引到内部点，且强凸性似乎没有向外运动的空间。我们通过$\rho=0.3$的显式、完全经认证的构造证明该期望是错误的：一个在主对数图上二次型、非对角耦合为0.7的代价函数，其测地Hessian在整个$\boldsymbol{\text{mathcal{C}}}$上满足$\text{Hess }f\boldsymbol{\text{succeq}}\boldsymbol{\text{μ}}I_3$，其中机器认证的模$\boldsymbol{\text{μ}}\boldsymbol{\text{≥}}0.172$，采用精确有理输入的严格球算术，但其边界点处的下降速度具有精确有理的向外径向分量$21/500$。精确速率的连续性推论证明该流会离开球；数值积分显示在收敛到极小值点前的峰值偏移接近$0.3143$。机制很基础：强凸性约束梯度在极小值方向上的投影，而非在内径向方向上的投影。该附注附带了可复现所有认证常数和图的代码。
英文摘要
Let $mathcal{C}=\overline{\mathcal{B}}_ρ(R_c)$ be a geodesic ball of radius $ρ<π/2$ in SO(3) with the bi-invariant metric, and let $f$ be geodesically strongly convex on $\mathcal{C}$ with an interior minimizer. It is tempting to expect the gradient flow $\dot R=R(-\nabla f)^\wedge$ to keep $\mathcal{C}$ forward invariant: the flow is attracted to an interior point, and strong convexity appears to leave no room for outward motion. We show this expectation is false by an explicit, fully certified construction with $ρ=0.3$: a cost, quadratic in the principal logarithmic chart with off-diagonal coupling $0.7$, whose geodesic Hessian satisfies $\Hess f\succeqμI_3$ on all of $\mathcal{C}$ with a machine-certified modulus $μ\geq0.172$, rigorous ball arithmetic over exact rational inputs, yet whose descent velocity at a boundary point has the exact rational outward radial component $21/500$. A continuity corollary of the exact rate certifies that the flow exits the ball; numerical integration puts the peak excursion near $0.3143$ before convergence to the minimizer. The mechanism is elementary: strong convexity constrains the projection of the gradient onto the minimizer direction, not onto the inward radial direction. Code reproducing every certified constant and figure accompanies the note.
