最小方差调和矩阵指数分布
Least-Variable Harmonic Matrix-Exponential Distributions
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中文总结 AI 辅助
本文提出一类更广的调和矩阵指数分布，将系数优化化为特征值问题，在保持最优下确界的同时，以更少参数获得更小SCV，并给出$O(N^{-2})$的上界。
中文摘要 AI 辅助
集中矩阵指数（CME）分布是近似固定时间的随机时钟，其质量由变异系数的平方（SCV）衡量；一个众所周知的构造方法是在共享指数阻尼下由余弦平方项的乘积构建此类时钟，但这些项的数量，以及需要优化的参数数量，随阶数增长。由于在高阶时对完整参数空间进行穷举搜索变得极其昂贵，先前的工作依赖于低维启发式参数化而非真正的最优解。我们引入了一类严格更大的公共阻尼调和密度类，由任意非负三角多项式而非此类乘积构建。对于每对固定的标量参数，系数优化简化为单个特征值问题，留下一个与阶数无关的二维非线性搜索。这种扩展使得下确界与经典余弦平方构造的下确界保持不变。在三个参数启发式允许比较的阶数上应用这一精确刻画，给出了更小的报告SCV值，且在更高阶时增益更大。另外，一个显式调和构造给出了可达到SCV的$O(N^{-2})$上界，其中$N$是ME表示预算，而Erlang分布的线性速率为$1/N$。
英文摘要
Concentrated matrix-exponential (CME) distributions are random clocks approximating a fixed time, with quality measured by the squared coefficient of variation (SCV); a well-known construction builds such clocks from products of cosine-squared terms under shared exponential damping, but the number of these terms, and hence the number of parameters to optimize, grows with the order. Since exhaustive search over the full parameter space becomes prohibitively expensive at high orders, previous work has relied on low-dimensional heuristic parametrizations rather than the true optimum. We introduce a strictly larger class of common-damping harmonic densities, built from an arbitrary nonnegative trigonometric polynomial rather than such a product. For each fixed pair of scalar parameters, coefficient optimization reduces to a single eigenvalue problem, leaving a two-dimensional nonlinear search independent of the order. The enlargement leaves the infimum unchanged from that of the classical cosine-squared construction. Applying this exact characterization at the orders where the three-parameter heuristic allows comparison gives smaller reported SCV values, with larger gains at higher orders. Separately, an explicit harmonic construction gives an $O(N^{-2})$ upper bound on the attainable SCV, where $N$ is the ME representation budget, compared with the Erlang distribution's linear rate $1/N$.
