Exploiting locality in high-dimensional factorial hidden Markov models
This work addresses computational bottlenecks in high-dimensional state-space models for applications with spatial or network structure, such as passenger flow analysis, representing an incremental improvement through principled approximation.
The authors tackled the problem of exponential computational cost in high-dimensional Factorial hidden Markov models by proposing approximate filtering and smoothing algorithms that discard likelihood factors based on locality in a factor graph, achieving dimension-free error bounds as proven in their analysis.
We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-dimension cost of exact filtering and smoothing to be avoided. We prove that the approximation accuracy, measured in a local total variation norm, is "dimension-free" in the sense that as the overall dimension of the model increases the error bounds we derive do not necessarily degrade. A key step in the analysis is to quantify the error introduced by localizing the likelihood function in a Bayes' rule update. The factorial structure of the likelihood function which we exploit arises naturally when data have known spatial or network structure. We demonstrate the new algorithms on synthetic examples and a London Underground passenger flow problem, where the factor graph is effectively given by the train network.