Sampling Over Riemannian Manifolds Using Kernel Herding
Sampling Over Riemannian Manifolds Using Kernel Herding
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DOI：
10.1109/icra46639.2022.9811951
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
Sandesh Adhikary;Byron Boots
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文献类型：
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作者：
Sandesh Adhikary;Byron Boots
核羊群是一种确定性抽样算法，当概率分布的核均值嵌入到再生核希尔伯特空间(RKHS)中时，该算法被设计来从概率分布中提取超级样本。使用这些超级样本形成的RKHS中的函数的经验预期往往比来自真实分布本身的随机抽样收敛得更快。核羊群的标准实现被限制在平坦的欧几里德空间上进行采样，这对于机器人等可能适合更一般的黎曼流形的应用来说并不理想。我们建议通过(1)使用包含流形的适当距离度量的几何感知核和(2)使用黎曼优化来约束羊群样本位于流形上，从而使核羊群适合于黎曼流形。我们评估了我们的方法在涉及机器人中常用的各种流形的问题，包括旋转矩阵的SO(3)流形、用于编码单位四元数的球面流形和对称正定矩阵的流形。我们证明，我们的方法在从加权粒子的经验分布进行重采样的任务上优于现有的替代方法，这是在粒子过滤等应用中遇到的问题。我们还演示了如何将黎曼核羊群作为核递归近似贝叶斯计算算法的一部分来估计黑盒模拟器的参数，包括灵巧机器人手模拟器的惯性矩阵。我们的结果证实，通过我们的核羊群方法利用几何信息比其他方法产生更好的结果，包括使用启发式投影的标准核羊群。
Kernel herding is a deterministic sampling algorithm designed to draw ‘super samples' from probability distributions when provided with their kernel mean embeddings in a reproducing kernel Hilbert space (RKHS). Empirical expectations of functions in the RKHS formed using these super samples tend to converge even faster than random sampling from the true distribution itself. Standard implementations of kernel herding have been restricted to sampling over flat Euclidean spaces, which is not ideal for applications such as robotics where more general Riemannian manifolds may be appropriate. We propose to adapt kernel herding to Riemannian manifolds by (1) using geometry-aware kernels that incorporate the appropriate distance metric for the manifold and (2) using Riemannian optimization to constrain herded samples to lie on the manifold. We evaluate our approach on problems involving various manifolds commonly used in robotics including the SO(3) manifold of rotation matrices, the spherical manifold used to encode unit quaternions, and the manifold of symmetric positive definite matrices. We demonstrate that our approach outperforms existing alternatives on the task of resampling from empirical distributions of weighted particles, a problem encountered in applications such as particle filtering. We also demonstrate how Riemannian kernel herding can be used as part of the kernel recursive approximate Bayesian computation algorithm to estimate parameters of black-box simulators, including inertia matrices of an Adroit robot hand simulator. Our results confirm that exploiting geometric information through our approach to kernel herding yields better results than alternatives including standard kernel herding with heuristic projections.