Banded matrices can be used as precision matrices in several models including linear state-space models, some Gaussian processes, and Gaussian Markov random fields. The aim of the paper is to make modern inference methods (such as variational inference or gradient-based sampling) available for Gaussian models with banded precision. We show that this can efficiently be achieved by equipping an automatic differentiation framework, such as TensorFlow or PyTorch, with some linear algebra operators dedicated to banded matrices. This paper studies the algorithmic aspects of the required operators, details their reverse-mode derivatives, and show that their complexity is linear in the number of observations.
翻译:在包括线性状态空间模型、一些高山进程和高西亚马可夫随机字段在内的若干模型中,可使用带宽矩阵作为精确矩阵。本文的目的是以带宽精确度为高斯模型提供现代推论方法(如变式推论或梯度抽样),我们表明,通过装备自动区分框架,如TensorFlow或PyTorrch,以及专门为带宽矩阵设置的线性代数操作员,可以有效地实现这一目标。本文研究所需操作者的算法方面,详细介绍其反向模式衍生物,并表明其复杂性在观测数量上是线性的。