Towards Flexibility and Interpretability of Gaussian Process State-Space Model

Gaussian process state-space model (GPSSM) has attracted much attention over the past decade. However, the model representation power of GPSSM is far from satisfactory. Most GPSSM works rely on the standard Gaussian process (GP) with a preliminary kernel, such as squared exponential (SE) kernel and Mat\'{e}rn kernel, which limit the model representation power and its application in complex scenarios. To address this issue, this paper proposes a novel class of probabilistic state-space model named TGPSSM that enriches the GP priors in the standard GPSSM through parametric normalizing flow, making the state-space model more flexible and expressive. In addition, by inheriting the advantages of sparse representation of GP models, we propose a scalable and interpretable variational learning algorithm to learn the TGPSSM and infer the latent dynamics simultaneously. By integrating a constrained optimization framework and explicitly constructing a non-Gaussian state variational distribution, the proposed learning algorithm enables the TGPSSM to significantly improve the capabilities of state space representation and model inference. Experimental results based on various synthetic and real datasets corroborate that the proposed TGPSSM yields superior learning and inference performance compared to several state-of-the-art methods. The accompanying source code is available at https://github.com/zhidilin/TGPSSM

Markovian Gaussian Process Variational Autoencoders

Deep generative models are widely used for modelling high-dimensional time series, such as video animations, audio and climate data. Sequential variational autoencoders have been successfully considered for many applications, with many variant models relying on discrete-time methods and recurrent neural networks (RNNs). On the other hand, continuous-time methods have recently gained attraction, especially in the context of irregularly-sampled time series, where they can better handle the data than discrete-time methods. One such class are Gaussian process variational autoencoders (GPVAEs), where the VAE prior is set as a Gaussian process (GPs), allowing inductive biases to be explicitly encoded via the kernel function and interpretability of the latent space. However, a major limitation of GPVAEs is that it inherits the same cubic computational cost as GPs. In this work, we leverage the equivalent discrete state space representation of Markovian GPs to enable a linear-time GP solver via Kalman filtering and smoothing. We show via corrupt and missing frames tasks that our method performs favourably, especially on the latter where it outperforms RNN-based models.Comment: Non-archival paper presented at Workshop on Continuous Time Methods for Machine Learning. The 39th International Conference on Machine Learning, Baltimor