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.

翻译：在输入数据是连续数据时作出预测和量化其不确定性是一项基本的学习挑战,最近引起越来越多的注意。我们开发了SigGPDE,这是高山进程(GPs)在顺序数据方面的一个新的可缩放的稀多推论框架。我们的贡献是双重的。首先,我们构建了稀疏近点的诱因变量,从而使由此产生的证据约束较低(ELBO)不要求任何矩阵倒置。第二,我们显示GP签字内核的梯度是双曲部分偏差方程(PDE)的解决方案。这一理论洞察使我们能够建立一个高效的反向分析算法,优化ELBO。我们展示了SigGPDE相对于现有方法的重大计算收益,同时在高达100万个多变时间序列的大型数据集上实现最先进的分类任务业绩。