Geometric Science of Information - 4th International Conference, GSI 2019, Toulouse, France, August 27-29, 2019, Proceedings
Geometric Science of Information - 4th International Conference, GSI 2019, Toulouse, France, August 27-29, 2019, Proceedings
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信息几何科学 - 第四届国际会议，GSI 2019，法国图卢兹，2019 年 8 月 27-29 日，会议记录
DOI：
10.1007/978-3-030-26980-7_69
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Barp A
中科院分区：
文献类型：
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作者：
Barp A
本文证明了[20]中利用辛结构构造的紧李群的Hamilton Monte Carlo方法可以从具有双不变度量的正则几何力学中恢复出来。因此，我们得到的李群上的哈密顿力学的各种配方之间的对应关系，其诱导HMC算法。我们恢复了测地线运动的欧拉-阿诺德公式，并通过选择具有适当不变性的度量来构造显式HMC方案，将[20，21]扩展到非紧李群。最后，我们解释了如何在齐性空间上的力学可以作为一个约束系统在其相关的李群。在某些重要的情况下，这些约束可以自然地由哈密顿量的对称性来处理。
In this paper we show that the Hamiltonian Monte Carlo method for compact Lie groups constructed in [20] using a symplectic structure can be recovered from canonical geometric mechanics with a bi-invariant metric. Hence we obtain the correspondence between the various formulations of Hamiltonian mechanics on Lie groups, and their induced HMC algorithms. Working onwe recover the Euler-Arnold formulation of geodesic motion, and construct explicit HMC schemes that extend [20, 21] to non-compact Lie groups by choosing metrics with appropriate invariances. Finally we explain how mechanics on homogeneous spaces can be formulated as a constrained system over their associated Lie groups. In some important cases the constraints can be naturally handled by the symmetries of the Hamiltonian.