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基础问题，在科学与工程等众多领域具有广泛应用。大多数逼近方法的关键挑战在于需要处理用于建模数据的参数化分布中难以计算的归一化常数。本文提出了一种新颖的李群上的斯坦因算子，由此衍生出的核斯坦因差异（KSD）是一种无需归一化的损失函数。我们给出了多项理论结果，刻画了这一新型李群核斯坦因差异的性质及其最小化器——即最小KSD估计量（MKSDE）的特性。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von Mises-Fisher分布的MKSDE闭式解。最后，我们通过实验证据展示了最小化KSD相较于极大似然估计的优势。