等价布朗再生核希尔伯特空间表示的优化几何
Optimization Geometry of Equivalent Brownian RKHS Representations
- Free University of Bozen-Bolzano（博尔扎诺自由大学）
- Universitat de València（瓦伦西亚大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究通过有限布朗再生核希尔伯特空间中的节点、增量和谱坐标，揭示了等价参数化下优化轨迹与条件数的坐标依赖效应，并验证了相关理论预测。
中文摘要 AI 辅助
等价的有限参数化可以表示相同的函数和内在范数，却诱导出不同的优化算法。我们在一个受控的有限布朗再生核希尔伯特空间中研究这一效应，该空间采用节点坐标、增量坐标和谱坐标。经典有限元、再生核希尔伯特空间插值、布朗协方差和混合边界离散余弦变换恒等式使共享假设类、布朗能量、逼近算子和坐标映射变得明确。我们的主要结果涉及这一固定模型的优化几何。在映射初始化、相同标量步长和相同小批量的条件下，节点和谱的梯度下降/随机梯度下降具有完全相同的映射轨迹。增量梯度下降是常数布朗/索伯列夫度量下的显式欧拉步，因子为$1/h$。对于布朗正则化最小二乘，$\u03ba_2(\mathbf H_{\mathrm{inc}})\le1+A/\rho$，在固定$A$、$\rho>0$及所述归一化下，与网格分辨率$G$无关。在所述标准Adam约定下，通用正交等变群恰好是符号置换；块离散余弦变换VIII变换不是其中之一。Float64测试在五个网格上数值验证了有限恒等式、映射单层和递归轨迹、条件数预测以及与定理匹配的Adam分离。因此，在不改变所表示函数、内在正则化器或逼近空间的情况下，坐标效应被孤立出来。
英文摘要
Equivalent finite parameterizations can represent the same functions and intrinsic norm yet induce different optimization algorithms. We study this effect in a controlled finite Brownian RKHS with nodal, increment, and spectral coordinates. Classical finite-element, RKHS-interpolation, Brownian-covariance, and mixed-boundary DCT identities make the shared hypothesis class, Brownian energy, approximation operator, and coordinate maps explicit. Our main results concern the optimization geometry of this fixed model. With mapped initialization, identical scalar steps, and identical minibatches, nodal and spectral GD/SGD have exactly the same mapped trajectories. Increment GD is an explicit Euler step for the constant Brownian/Sobolev metric, with factor $1/h$. For Brownian-regularized least squares, $κ_2(\mathbf H_{\mathrm{inc}})\le1+A/ρ$, independently of grid resolution $G$ for fixed $A$, $ρ>0$, and the stated normalization. Under the stated standard-Adam convention, the universal orthogonal equivariance group is exactly the signed permutations; the block DCT-VIII transform is not one. Float64 tests over five grids numerically verify the finite identities, mapped one-layer and recursive trajectories, conditioning predictions, and theorem-matched Adam separation. Thus coordinate effects are isolated without changing the represented functions, intrinsic regularizer, or approximation space.
