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 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, 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 rich class of SDEs, we achieve competitive or even state-of-the-art results on various time series interpolation/classification problems.

翻译：我们考虑在潜变量模型中进行变分贝叶斯推断的问题，其中（可能复杂的）观测随机过程由潜在随机微分方程的解所支配。受学习（几乎任意的）潜在神经随机微分方程时面临的挑战（如高效梯度计算）启发，我们退一步研究一个特定子类。在我们的设置中，随机微分方程在齐性潜在空间上演化，并由对应（矩阵）李群的随机动力学诱导。在学习问题中，单位n维球面上的随机微分方程无疑是该设置最相关的具体形式。值得注意的是，对于变分推断，球面不仅便于使用真正无信息先验，而且我们在证据下界中获得了近似后验与先验过程之间KL散度的特别简洁直观的表达式。实验表明，所提出类型的潜在随机微分方程可通过现有的单步几何Euler-Maruyama方案高效学习。尽管将自身限制于较不丰富的随机微分方程类别，我们在各种时间序列插值/分类问题上仍取得了具有竞争力甚至最先进的结果。