发表机构
École Polytechnique Fédérale de Lausanne; KAUST（洛桑联邦理工学院; 阿卜杜拉国王科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究主动子空间方法的样本复杂度，通过投影误差度量与正则化逆 Christoffel 函数推导非渐近界，并利用再生核希尔伯特空间光滑性改进估计，从而全面表征先验样本复杂度。
AI 中文摘要
主动子空间通过估计梯度协方差算子的主特征空间，识别高维参数到输出映射中的低维线性结构。在实践中，该协方差由基于有限次数梯度评估的蒙特卡洛估计器替代。基于算子范数控制协方差误差的经典分析，得出的样本复杂度估计可能比计算中常用的采样规则悲观得多。本文直接在脊近似相关的投影误差度量下研究经验主动子空间方法。我们推导了由与梯度场相关的正则化逆 Christoffel 函数控制的非渐近拟最优性界。在有界梯度假设下，所得估计已经改进了从算子范数协方差界获得的样本复杂度估计。然后我们表明，梯度映射的额外光滑性（通过再生核希尔伯特空间中的成员资格表达）产生更锐利的相干性估计，并激发从核对角测度进行可处理的的重要性采样。此外，相同的光滑性假设为总体主动子空间尾部能量提供了先验衰减界，可与我们的有限样本估计相结合，以规定秩、正则化尺度和样本大小，从而完全表征先验样本复杂度。对于对数正态高斯和仿射均匀参数椭圆型偏微分方程，使用 Hermite 和 Legendre 级数展开的加权可和性验证了抽象假设。
英文摘要
Active subspaces identify low-dimensional linear structure in high-dimensional parameter-to-output maps by estimating the dominant eigenspace of a gradient covariance operator. In practice this covariance is replaced by a Monte Carlo estimator built from a limited number of gradient evaluations. Classical analyses based on controlling the covariance error in operator norm lead to sample-complexity estimates that can be substantially more pessimistic than the sampling rules commonly used in computations. This paper studies the empirical active subspace method directly in the projection-error metric relevant for ridge approximation. We derive non-asymptotic quasi-optimality bounds governed by a regularized inverse Christoffel function associated with the gradient field. Under a bounded-gradient assumption, the resulting estimates already improve the sample-complexity estimates obtained from operator-norm covariance bounds. We then show that additional smoothness of the gradient map, expressed through membership in a reproducing kernel Hilbert space, yields sharper coherence estimates and motivates tractable importance sampling from kernel diagonal measures. Furthermore, the same smoothness assumption yields a priori decay bounds for the population active subspace tail energy, which can be combined with our finite-sample estimate to prescribe rank, regularization scale, and sample size, allowing to fully characterize the a priori sample complexity. The abstract assumptions are verified for lognormal Gaussian and affine uniform parametric elliptic PDEs using weighted summability of Hermite and Legendre series expansions.
