为流形神经网络设计了考虑曲率的优化方法，提升训练稳定性与收敛性。
MSINO: Curvature-Aware Sobolev Optimization for Manifold Neural Networks
- 用协变Sobolev损失替代欧氏导数监督，结合平行传输对齐梯度。
- 理论证明梯度下降在曲率受限区域可线性收敛，步长有明确上限。
- 适用于三维旋转、机器人运动等流形学习场景，支持理论保障。
我们提出流形Sobolev感知神经优化（MSINO），一种针对定义在黎曼流形上的神经网络的曲率感知训练框架。该方法将标准欧氏导数监督替换为协变Sobolev损失，通过平行传输对齐梯度，并引入拉普拉斯-贝尔特拉米平滑正则项提升稳定性。基于黎曼优化与流形上Sobolev理论的经典结果，我们推导出几何依赖常数，实现：(i) 带流形Sobolev光滑常数的下降引理；(ii) Sobolev Polyak-Łojasiewicz不等式，给出黎曼梯度下降与随机梯度下降在线性收敛下的显式步长约束；(iii) 两步牛顿Sobolev方法，在曲率控制邻域内具有局部二次收缩。与欧氏空间中以往的Sobolev训练不同，MSINO提供显式追踪曲率与传输雅可比的训练时间保证。应用涵盖表面成像、物理信息学习以及在SO(3)和SE(3)等李群上的机器人任务。该框架统一了值函数与梯度驱动学习，为流形上神经网络训练提供曲率感知的收敛保障。
原文摘要 · Abstract (English)
We introduce Manifold Sobolev Informed Neural Optimization (MSINO), a curvature aware training framework for neural networks defined on Riemannian manifolds. The method replaces standard Euclidean derivative supervision with a covariant Sobolev loss that aligns gradients using parallel transport and improves stability via a Laplace Beltrami smoothness regularization term. Building on classical results in Riemannian optimization and Sobolev theory on manifolds, we derive geometry dependent constants that yield (i) a Descent Lemma with a manifold Sobolev smoothness constant, (ii) a Sobolev Polyak Lojasiewicz inequality giving linear convergence guarantees for Riemannian gradient descent and stochastic gradient descent under explicit step size bounds, and (iii) a two step Newton Sobolev method with local quadratic contraction in curvature controlled neighborhoods. Unlike prior Sobolev training in Euclidean space, MSINO provides training time guarantees that explicitly track curvature and transported Jacobians. Applications include surface imaging, physics informed learning settings, and robotics on Lie groups such as SO(3) and SE(3). The framework unifies value and gradient based learning with curvature aware convergence guarantees for neural training on manifolds.
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