布朗核梯子的公共协方差几何与认证

中文摘要 AI 辅助

一种表示自适应的核类在固定样本上产生再生核希尔伯特空间椭球的并集，而非单个椭球。我们引入了最小迹公共协方差，它支配由布朗核梯子生成的无约束经验并集，并发展了其统计、逼近理论和计算方面的后果。协方差值通过绝对二求和算子和协方差支配乘子具有精确表述，并产生一个通用的高斯复杂度界。一个封闭的末层狄拉克迹约简和符号布朗阈值表示将通用协方差问题转化为阈值、图余面积和有效电阻几何。这些工具给出了确定性深度定律、条件高斯逆、随机设计和扰动转移，以及一个精确的经验科尔莫戈罗夫宽度公式，其主协方差特征空间同时逼近完整的自适应球。有限接触、活动半定规划、验证分离和凸电阻设计松弛提供了互补的下界和上界证书。在冻结表示上的有限协方差索引布朗路径说明了成功协方差认证与预测选择之间的区别：所有报告的路径证书均成功，而锁定的预测研究错过了一个预先声明的聚合标准。因此，本文确定了一个连接无约束核自适应、高斯几何、公共子空间和可认证计算的有限维协方差对象。

英文摘要

A representation-adaptive kernel class produces, on a fixed sample, a union of reproducing-kernel Hilbert-space ellipsoids rather than one ellipsoid. We introduce the minimum-trace common covariance that dominates the unrestricted empirical union generated by Brownian kernel ladders and develop its statistical, approximation-theoretic, and computational consequences. The covariance value admits exact formulations through absolutely two-summing operators and covariance-dominated multipliers, and it yields a universal Gaussian-complexity bound. A closed last-layer Dirac-trace reduction and a signed Brownian threshold representation convert the generic covariance problem into threshold, graph-coarea, and effective-resistance geometry. These tools give deterministic depth laws, conditional Gaussian reverses, random-design and perturbation transfers, and an exact empirical Kolmogorov-width formula whose leading covariance eigenspaces approximate the complete adaptive ball simultaneously. Finite contact, active semidefinite programs, verified separation, and a convex resistance-design relaxation provide complementary lower and upper certificates. A finite covariance-indexed Brownian path on frozen representations illustrates the distinction between successful covariance certification and predictive selection: all reported path certificates succeed, whereas the locked predictive study misses one predeclared aggregate criterion. The paper thereby identifies one finite-dimensional covariance object linking unrestricted kernel adaptation, Gaussian geometry, common subspaces, and certifiable computation.

发表机构

• Free University of Bozen–Bolzano（博尔扎诺自由大学）

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