发表机构
Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
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Chebyshev精确加速递推在Hessian扰动下的最优性，证明sine-Jacobi递推渐近最优，并通过计算机辅助变分方法给出严格下界。
AI 中文摘要
我们研究具有正步长和非负动量系数的两步递推，其终端残差为指定谱区间$[\mu,L]$上的Chebyshev极小极大多项式。这些递推在固定二次函数上具有相同的终端误差，但它们对逐步Hessian扰动的一阶响应可能不同。我们证明，第一部分中的sine-Jacobi递推在该响应范数意义下渐近最优，误差因子为$1.000004$，且对$0<\mu<L$一致成立。每个终端精确递推满足$A_N\ge(2\sqrt{c_{cert}}-o(1))N^{3/2}\epsilon_N^\star/(L-\mu)$，其中$c_{cert}>1.14268353624$，而sine-Jacobi递推达到常数$2\sqrt{c_{\sin}}$，其中$c_{\sin}<1.14269218$。证明将增益归结为Jacobi Green矩阵的对角能量。一个中心化提升将该矩阵表示为指数核；将其对角线打包为直线上的密度，保持能量并控制Birman-Schwinger谱的变化。一个校准恒等式利用基态参数和一个Fredholm行列式给出下界。另外两个行列式和一个迹观测通过显式紧致状态区域上的逐点不等式改进该下界。一个有理证书通过区间算术验证该不等式。第二个证书构造一个密度，满足所选观测和两个归一化条件，且能量低于sine值。最后，在指定类型的有限多个观测中，不存在$C^1$函数能在sine密度处给出局部尖锐下界。
英文摘要
We study two-step recurrences with positive step sizes and nonnegative momentum coefficients whose terminal residual is the Chebyshev minimax polynomial on a prescribed spectral interval $[μ,L]$. These recurrences have the same terminal error on a fixed quadratic, but their first-order responses to stepwise Hessian perturbations can differ. We prove that the sine-Jacobi recurrence of Part I is asymptotically optimal for this response norm within a factor of $1.000004$, uniformly over $0<μ<L$. Every terminal-exact recurrence satisfies $A_N\ge(2\sqrt{c_{cert}}-o(1))N^{3/2}ε_N^\star/(L-μ)$ with $c_{cert}>1.14268353624$, whereas the sine-Jacobi recurrence attains the constant $2\sqrt{c_{\sin}}$ with $c_{\sin}<1.14269218$. The proof reduces the gain to the diagonal energy of a Jacobi Green matrix. A centering lift represents this matrix by an exponential kernel; packing its diagonal into a density on the line preserves the energy and controls the change in the Birman-Schwinger spectrum. A calibrated identity gives a lower bound using the ground parameter and one Fredholm determinant. Two further determinants and a trace observation improve the bound through a pointwise inequality on an explicit compact state region. One rational certificate verifies this inequality by interval arithmetic. A second certificate constructs a density satisfying the selected observations and both normalizations with energy below the sine value. Finally, no $C^1$ function of finitely many observations of the specified kinds gives a locally sharp lower bound at the sine density.
Comments64 pages
