Title: Scalable Continuous-Time Hidden Markov Models
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Hidden Markov Model; Forward Algorithm; continuous-time

Abstract: As a probabilistic tool for irregularly-sampled data, the Continuous-Time Hidden Markov Model (CTHMM) inherently handles real phenomena with uncertainties modelled by distributions. However, CTHMM is affected by (i) the costly matrix exponentiation (cubic time complexity \wrt the number of hidden states) involved in the estimation of transition probabilities, and (ii) the use of simplistic parametric observation models (e.g., Gaussian). Thus, we propose scalable algorithms for CTHMM on traditional problems (learning, evaluation, decoding) to ensure tractability. Firstly, we factorise states of CTHMM into multiple binary states (e.g., several $2\times 2$ sub-problems) leading to a distributed closed-form exponentiation. We also accelerate matrix-vector products, reducing the complexity from quadratic to linearithmic. Secondly, the simplistic parametric distributions are replaced by the normalising flows (that transform simple distributions into complex data-driven distributions), accelerated by sharing few invertible neural networks among groups of hidden states. Training our approach takes few hours on a GPU, while standard CTHMMs with mere 10 hidden states take few weeks.  On the largest dataset, our method scales favourably (up to 1024$\times$ larger hidden states than naive CTHMM and outperforms it by 4.1 in log-likelihood). We also outperform competing HMMs with advanced solvers on downstream tasks.

TL;DR: Make continuous-time Hidden Markov Models scalable.