Distributional approximation is a fundamental problem in machine learning with numerous applications across all fields of science and engineering and beyond. The key challenge in most approximation methods is the need to tackle the intractable normalization constant pertaining to the parametrized distributions used to model the data. In this paper, we present a novel Stein operator on Lie groups leading to a kernel Stein discrepancy (KSD) which is a normalization-free loss function. We present several theoretical results characterizing the properties of this new KSD on Lie groups and its minimizers namely, the minimum KSD estimator (MKSDE). Proof of several properties of MKSDE are presented, including strong consistency, CLT and a closed form of the MKSDE for the von Mises-Fisher distribution on SO(N). Finally, we present experimental evidence depicting advantages of minimizing KSD over maximum likelihood estimation.

翻译：分布逼近是机器学习中的一个基本问题，在科学与工程乃至更广泛领域具有众多应用。大多数逼近方法的关键挑战在于需要处理与用于建模数据的参数化分布相关的难以计算的归一化常数。本文提出了一种李群上的新型斯坦因算子，由此导出一个无需归一化的损失函数——核斯坦因差异。我们给出了关于该李群上新核斯坦因差异及其最小化器（即最小核斯坦因差异估计量）性质的若干理论结果。文中证明了最小核斯坦因差异估计量的多个性质，包括强相合性、中心极限定理，以及SO(N)上冯·米塞斯-费希尔分布的最小核斯坦因差异估计量闭式解。最后，我们通过实验证据展示了最小化核斯坦因差异相较于极大似然估计的优势。