Making predictions and quantifying their uncertainty when the input data is
sequential is a fundamental learning challenge, recently attracting increasing
attention. We develop SigGPDE, a new scalable sparse variational inference
framework for Gaussian Processes (GPs) on sequential data. Our contribution is
twofold. First, we construct inducing variables underpinning the sparse
approximation so that the resulting evidence lower bound (ELBO) does not
require any matrix inversion. Second, we show that the gradients of the GP
signature kernel are solutions of a hyperbolic partial differential equation
(PDE). This theoretical insight allows us to build an efficient
back-propagation algorithm to optimize the ELBO. We showcase the significant
computational gains of SigGPDE compared to existing methods, while achieving
state-of-the-art performance for classification tasks on large datasets of up
to 1 million multivariate time series.

本文我们提出了 SigGPDE，它是一个新的可扩展的稀疏变分推理框架，用于处理序列数据上的高斯过程。我们的贡献有两个方面：首先，我们构造了诱导变量来支撑稀疏近似，使得得到的证据下限 ELBO 不需要任何矩阵求逆；其次，我们展示了 GP 签名核的梯度是一个双曲型偏微分方程（PDE）的解，这一理论洞见使我们能够构建一个有效的反向传播算法来优化 ELBO。通过展示 SigGPDE 相对于现有方法的显着计算收益，同时在多达 1 百万个多元时间序列的大型数据集上实现最先进的分类任务性能。