Tensorized algorithms and scalable filtering methods for hidden Markov and factorial hidden Markov models

Factorial hidden Markov models (fHMMs) are powerful tools for modeling time-series data influenced by multiple independent latent factors. However, standard approaches typically reformulate an fHMM into a single, equivalent hidden Markov model (HMM). This transformation leads to a state-space explosion, making the forward filtering algorithm—essential for evaluation, decoding, and parameter estimation—computationally intractable for all but the smallest systems.

This paper proposes a novel, scalable filtering approach that bypasses the construction of intermediate HMM representations. Instead, the authors leverage multilinear algebra to directly manipulate the multidimensional structure of the fHMM. By treating the factorial subsystems as tensors, the authors reformulate the forward filtering steps using tensor contractions and products. This approach allows the model to maintain the independence of its constituent Markov chains throughout the computation, significantly reducing the memory and processing requirements compared to naive HMM-based implementations.

The study demonstrates that tensorized algorithms can efficiently perform forward filtering for systems with multiple subsystems and large state-spaces. By utilizing optimized tensor operations (such as those found in NumPy or MATLAB), the proposed method enables the analysis of complex, data-intensive systems that were previously restricted by the computational bottlenecks of traditional HMM formulations. The authors provide a rigorous mathematical framework for these tensorized operations, showing that they are mathematically equivalent to standard filtering but computationally superior.

As data-intensive applications in fields like bioinformatics, finance, and signal processing continue to grow, the ability to model systems with multiple interacting factors is increasingly important. This work provides a practical, scalable framework that allows researchers to apply fHMMs to larger, more realistic datasets without sacrificing model fidelity or incurring prohibitive computational costs.