Sliced Stein discrepancy (SSD) and its kernelized variants have demonstrated promising successes in goodness-of-fit tests and model learning in high dimensions. Despite their theoretical elegance, their empirical performance depends crucially on the search of optimal slicing directions to discriminate between two distributions. Unfortunately, previous gradient-based optimisation approaches for this task return sub-optimal results: they are computationally expensive, sensitive to initialization, and they lack theoretical guarantees for convergence. We address these issues in two steps. First, we provide theoretical results stating that the requirement of using optimal slicing directions in the kernelized version of SSD can be relaxed, validating the resulting discrepancy with finite random slicing directions. Second, given that good slicing directions are crucial for practical performance, we propose a fast algorithm for finding such slicing directions based on ideas of active sub-space construction and spectral decomposition. Experiments on goodness-of-fit tests and model learning show that our approach achieves both improved performance and faster convergence. Especially, we demonstrate a 14-80x speed-up in goodness-of-fit tests when comparing with gradient-based alternatives.

翻译：切片斯坦差异(SSD)及其内核变体在高层次的完善测试和模型学习中表现出了有希望的成功。尽管它们的理论优雅度,但它们的经验性表现关键取决于对最佳切片方向的搜索,以区分两种分布。不幸的是,以往对这项任务的梯度优化方法返回了亚最佳结果:它们计算成本昂贵,对初始化敏感,缺乏理论保证。我们分两个步骤处理这些问题。首先,我们提供理论结果,说明在精密的SSD版本中使用最佳切片方向的要求可以放松,从而证实由此产生的差异与有限的随机切片方向的不一致。第二,鉴于良好的切片方向对实际绩效至关重要,我们提出一种快速算法,以根据活跃的子空间构造和光谱分解的理念来找到这种切片方向。关于优美测试和模型学习的实验表明,我们的方法既提高了性能,也加快了趋同速度。特别是,在与基于梯度的替代品进行比较时,我们展示了14-80x的精美度测试速度。