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：我们考虑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）的解所支配。受学习大规模数据中（几乎任意的）潜神经SDE时面临的挑战（如高效梯度计算）启发，我们退一步转而研究特定子类方法。在此框架下，SDE在齐性潜空间上演化，并由对应（矩阵）李群的随机动力学驱动。在学习问题中，单位$n$维球面上的SDE是此类设定中最具代表性的实例。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先验SDE，而且我们还能获得关于证据下界中近似后验与先验过程之间的KL散度的特别简洁直观的表达式。实验表明，通过现有单步几何欧拉-丸山格式可高效学习所提类型的潜SDE。尽管将SDE限制在较少的类别中，我们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。