本文提出了一个用于逼近一类次指数过程上确界的算法,具体来说,给定一个有限的向量集合V⊆ℝd,对于集合上密度函数对称单峰的次指数过程X,我们能够在多项式时间内确定性地计算出其上确界的期望,即E[ supv∈V| 〈 v,X 〉 | ]的(1+ε)阶的近似...本文提出了一个用于逼近一类次指数过程上确界的算法,具体来说,给定一个有限的向量集合V⊆ℝd,对于集合上密度函数对称单峰的次指数过程X,我们能够在多项式时间内确定性地计算出其上确界的期望,即E[ supv∈V| 〈 v,X 〉 | ]的(1+ε)阶的近似值,其中X服从d维正态分布,ε是一个大于0的常数。在此前,相关的工作只研究了高斯过程的上确界的算法,而次指数过程作为高斯过程的扩展,在泛函分析、凸几何以及有限图上的随机游走等领域有着广泛的应用,其上确界的近似算法在高斯假设过强的场景下具有重要的研究价值,可以提供的合理的理论保证。This paper proposes an algorithm for approximating the upper bound of a class of sub-exponential processes. Specifically, given a finite set of vectors V⊆ℝd, for a sub-exponential process X with a density function that is symmetric and unimodal on the set, we can deterministically compute the expected upper bound in polynomial time, that is, the (1+ε)-th order approximation of EX←Nd[ supv∈V| 〈 v,X 〉 | ], where X follows a d-dimensional normal distribution, and εis a constant greater than 0. Prior to this, related work has only studied algorithms for the upper bounds of Gaussian processes, while sub-exponential processes, as an extension of Gaussian processes, have a wide range of applications in functional analysis, convex geometry, and random walks on finite graphs, among other fields. The approximation algorithm for the upper bound has significant research value in scenarios where the Gaussian assumption is too strong, providing a reasonable theoretical guarantee.展开更多
文摘液性指数是研究土壤稳定性、土体变形、土体强度等问题的关键参数,因此对液性指数的准确预测至关重要。基于南京和合肥地区黏性土的孔压静力触探(piezocone penetration test,简称CPTU)原位测试数据集,以室内液塑限试验计算的液性指数为参考值,采用支持向量回归(support vector regression,简称SVR)、粒子群算法优化支持向量回归(particle swarm optimization based SVR,简称PSO-SVR)、遗传算法优化支持向量回归(genetic algorithm based SVR,简称GA-SVR)、模拟退火算法优化支持向量回归(simulated annealing based SVR,简称SA-SVR)对土体的液性指数进行评价,并将预测结果与室内试验结果以及CPTU经验公式对比。为更贴近工程实践,以原位测试时的孔洞为单位,进行单孔预测分析,最后,进行参数敏感性分析。结果表明,SVR模型和优化的SVR模型,都能预测黏性土的液性指数,算法优化后的3种模型在性能上表现更好。单孔分析时,SA-SVR模型以波动平滑、峰值适中等优点,预测效果更佳。工程实践中,建议采用归一化锥尖阻力、摩阻比、孔压参数比、上覆应力及有效上覆应力作为输入变量。PSO-SVR模型、GA-SVR模型、SA-SVR模型敏感性走向均与理论相同,但SA-SVR模型跨度更小,与理论结果更加一致,验证了SA-SVR模型的准确性。所提出的SA-SVR模型可以更好地预测黏性土的液性指数,并指导工程实践。
文摘本文提出了一个用于逼近一类次指数过程上确界的算法,具体来说,给定一个有限的向量集合V⊆ℝd,对于集合上密度函数对称单峰的次指数过程X,我们能够在多项式时间内确定性地计算出其上确界的期望,即E[ supv∈V| 〈 v,X 〉 | ]的(1+ε)阶的近似值,其中X服从d维正态分布,ε是一个大于0的常数。在此前,相关的工作只研究了高斯过程的上确界的算法,而次指数过程作为高斯过程的扩展,在泛函分析、凸几何以及有限图上的随机游走等领域有着广泛的应用,其上确界的近似算法在高斯假设过强的场景下具有重要的研究价值,可以提供的合理的理论保证。This paper proposes an algorithm for approximating the upper bound of a class of sub-exponential processes. Specifically, given a finite set of vectors V⊆ℝd, for a sub-exponential process X with a density function that is symmetric and unimodal on the set, we can deterministically compute the expected upper bound in polynomial time, that is, the (1+ε)-th order approximation of EX←Nd[ supv∈V| 〈 v,X 〉 | ], where X follows a d-dimensional normal distribution, and εis a constant greater than 0. Prior to this, related work has only studied algorithms for the upper bounds of Gaussian processes, while sub-exponential processes, as an extension of Gaussian processes, have a wide range of applications in functional analysis, convex geometry, and random walks on finite graphs, among other fields. The approximation algorithm for the upper bound has significant research value in scenarios where the Gaussian assumption is too strong, providing a reasonable theoretical guarantee.
