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、工程及其他领域的众多应用中均有体现。大多数逼近方法面临的主要挑战在于，需要处理用于建模数据的参数化分布所涉及的难解归一化常数。本文提出了一种基于李群的新型斯坦因算子，进而导出一种无需归一化的损失函数——核斯坦因偏差（KSD）。我们给出了若干理论结果，用以刻画这种新型KSD在李群上的性质及其极小化器，即最小KSD估计量（MKSDE）。我们证明了MKSDE的若干性质，包括强相合性、中心极限定理，以及SO(N)上冯·米塞斯-费希尔分布对应MKSDE的闭式解。最后，我们通过实验证据展示了最小化KSD相较于最大似然估计的优势。