AI 中文总结
研究维纳-埃尔米特互相关识别在非高斯激励下的失配惩罚，通过特定方法给出精确有限阶超额$L^2(P)$风险，利用封闭累积量形式揭示特征，自举法可区分不同信道，计算经机器检查。
AI 中文摘要
维纳-埃尔米特互相关识别表示埃尔米特基中的多项式响应。在高斯激励下，该基是正交的，对角规则能精确恢复它；在非高斯激励下，保持相同的基，但它的格拉姆矩阵有非对角项，对角规则不再是总体投影。我们给出了这种失配的精确有限阶超额$L^2(P)$风险：由两个汉克尔-乔列斯基分解和一次对角求解得到的矩二次型，从矩到2s阶的成本为$O(s^3)$。三阶和四阶的封闭累积量形式揭示了哪些非高斯特征驱动了它；对称性仅在二阶之前保护高斯基。一种自举法根据数据决定是否值得构建匹配基；在维纳-哈默斯坦基准测试中，它将近高斯信道（惩罚约为$10^{-4}$）与偏态输出（惩罚为0.05）区分开来。该计算是一种加权$L^2$投影，其核心正规系统对应关系在Lean 4中经过机器检查。
英文摘要
Wiener-Hermite cross-correlation identification represents a polynomial response in the Hermite basis. Under Gaussian excitation the basis is orthogonal and a diagonal rule recovers it exactly; under non-Gaussian excitation the same basis is kept, but its Gram matrix gains off-diagonal terms and the diagonal rule is no longer the population projection. We give the exact finite-order excess $L^2(P)$ risk of this mismatch: a moment quadratic form from two Hankel-Cholesky factorizations and one diagonal solve, at $O(s^3)$ cost from moments to order $2s$. Closed cumulant forms at orders three and four expose which non-Gaussian features drive it; symmetry protects the Gaussian basis only through order two. A bootstrap decides, from data, whether a matched basis is worth building; on a Wiener-Hammerstein benchmark it separates a near-Gaussian channel (penalty $\approx 10^{-4}$) from a skewed output (penalty $0.05$). The computation is a weighted-$L^2$ projection whose core normal-system correspondence is machine-checked in Lean 4.
Comments23 pages, 2 figures. Includes a Lean 4/Mathlib machine-checked core. Reproducibility code (MIT): https://github.com/SZabolotnii/Ku-Projection-Framework-code-supplement
