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。

SigGPDE：在序列数据上扩展稀疏高斯过程