We consider the problem of variational Bayesian inference in a latent variable model where a (possibly complex) observed stochastic process is governed by the solution of a latent stochastic differential equation (SDE). Motivated by the challenges that arise when trying to learn an (almost arbitrary) latent neural SDE from large-scale data, such as efficient gradient computation, we take a step back and study a specific subclass instead. In our case, the SDE evolves on a homogeneous latent space and is induced by stochastic dynamics of the corresponding (matrix) Lie group. In learning problems, SDEs on the unit $n$-sphere are arguably the most relevant incarnation of this setup. Notably, for variational inference, the sphere not only facilitates using a truly uninformative prior SDE, but we also obtain a particularly simple and intuitive expression for the Kullback-Leibler divergence between the approximate posterior and prior process in the evidence lower bound. Experiments demonstrate that a latent SDE of the proposed type can be learned efficiently by means of an existing one-step geometric Euler-Maruyama scheme. Despite restricting ourselves to a less diverse class of SDEs, we achieve competitive or even state-of-the-art performance on various time series interpolation and classification benchmarks.

翻译：我们研究潜变量模型中的变分贝叶斯推断问题，其中（可能复杂的）观测随机过程由潜随机微分方程（SDE）的解控制。受大规模数据下学习（近乎任意）潜神经SDE所面临的挑战（如高效梯度计算）启发，我们退而求其次，转而研究一个特定子类。在我们的设置中，SDE在齐性潜空间上演化，并由相应（矩阵）李群的随机动力学诱导。在学习问题中，单位$n$球面上的SDE堪称该框架最相关的实例。值得注意的是，对于变分推断而言，球面不仅便于使用真正无信息的先验SDE，我们还能在证据下界中获得近似后验过程与先验过程之间KL散度的极简直观表达式。实验表明，通过现有的单步几何欧拉-丸山格式可高效学习所提类型的潜SDE。尽管我们将自身限制于多样性较低的SDE类别，但在各类时间序列插值与分类基准测试中仍取得了具有竞争力乃至最先进的性能。