变分布朗核梯

中文摘要 AI 辅助

关于深度优势的主张取决于分配给表示的复杂度。我们引入变分布朗核梯（Variation Brownian Kernel Ladder，VBKL），这是一种路径原子型函数空间框架，将非线性递归字典构建与线性变分叠加分离开来。从线性投影出发，每个原子递归地由布朗再生核希尔伯特空间（RKHS）的单位球轮廓构成；完整的VBKL空间则是已完成字典的符号测度变分包络。我们将每个递归字典识别为布朗拉回RKHS球的并集，建立了变分控制的赫尔德正则性、紧性与可达性，以及在局部非退化条件下随深度严格增长的性质，该条件的迹位于输入测度的支撑集内。对于相关的有限低支撑架构，我们通过布朗二次混沌、符号阈值迹和VC维导出了拉德马赫（Rademacher）界与泛化界。我们还通过离散化外测度和选定的外布朗轮廓构造了两级近似，得到了M⁻¹/² + m⁻¹/²的误差界、尖锐插值常数√(A/2)，以及每次评估至多2M个活跃外轮廓基元贡献的结果。受控实验阐明了其近似机制，并表明在有限数据下具有良好的精度-复杂度权衡。

英文摘要

Claims about the benefit of depth depend on the complexity assigned to a representation. We introduce the \emph{Variation Brownian Kernel Ladder} (VBKL), a path-atomic function-space framework that separates nonlinear recursive dictionary construction from linear variation superposition. Starting from linear projections, each atom recursively composes unit-ball profiles from the Brownian reproducing kernel Hilbert space; the full VBKL space is then the signed-measure variation hull of the completed dictionary. We identify each recursive dictionary as a union of Brownian pullback RKHS balls and establish variation-controlled Hölder regularity, compactness and attainment, and strict growth with depth under a local non-degeneracy condition whose trace lies in the support of the input measure. For associated finite lower-support architectures, we derive Rademacher and generalization bounds through Brownian quadratic chaos, signed threshold traces, and VC entropy. We also construct two-stage approximants by discretizing the outer measure and the selected outer Brownian profiles, obtaining an $M^{-1/2}+m^{-1/2}$ error bound, a sharp interpolation constant $\sqrt{A/2}$, and at most $2M$ active outer-profile basis contributions per evaluation. Controlled experiments illustrate the approximation mechanisms and indicate a favorable limited-data accuracy--complexity trade-off.

发表机构

• Faculty of Engineering, Free University of Bozen-Bolzano（博尔扎诺自由大学工程学院）

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