We present three simulation schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces and use numerical results of the guided processes in the Lie group $\SO(3)$ and on the homogeneous spaces $\mathrm{SPD}(3) = \mathrm{GL}_+(3)/\mathrm{SO}(3)$ and $\mathbb S^2 = \mathrm{SO}(3)/\mathrm{SO}(2)$ to evaluate our sampling scheme. Brownian motions on Lie groups can be defined via the Laplace-Beltrami of a left- (or right-)invariant Riemannian metric. Given i.i.d. Lie group-valued samples on $\mathrm{SO}(3)$ drawn from a Brownian motion with unknown Riemannian metric structure, the underlying Riemannian metric on $\mathrm{SO}(3)$ is estimated using an iterative maximum likelihood (MLE) method. Furthermore, the re-sampling technique is applied to yield estimates of the heat kernel on the two-sphere considered as a homogeneous space. Comparing this estimate to the truncated version of the closed-form expression for the heat kernel on $\mathbb S^2$ serves as a proof of concept for the validity of the sampling scheme on homogeneous spaces.
翻译:我们提出了三个模拟方案,用以模拟完整和相连的谎言组和同质空间上的布朗桥,并使用利伊组中左(右)异性里伊曼尼测量值的数值结果。鉴于i.d.d.,从具有不为人知的里曼度结构的布朗运动中提取的利伊组估价样品美元(3)和美元马特布S ⁇ 2 = mathrm{SO}(3)/\mathrm{SO}(2)美元,用于评估我们的取样方案。对利伊组中的布朗运动可以通过利伊组中左(右)异性里伊曼尼测量度测量值的Laplace-Beltrami确定数字结果。根据i.d.d.,从利伊组估值样品美元-基数为美元,取自具有未知里曼度结构的布朗运动,对赖伊曼基标准值的瑞曼尼基标准值是使用一种迭代最大可能性(MLE)方法估算的。此外,再采样技术应用用于在两层平基内热内热内热层的热内层估计,即Smal-cal-cregreal strapreval sal ser sal syal sal sal sal servial sal sal supal sal sal sal sal supal sal sal sal sal sal sy sy syal sal sal sypal sy sal sypal sal sypal sal sal sal sal sal sypal supal sal sal sal sal sal sal sypal sypal sal sal sal sal sal sal supal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sal sipection. sipection. siction. siction. sipection. siction.