# Algebraic Reduction of Hidden Markov Models

arXiv: https://arxiv.org/abs/2208.05968 · HTML: https://mlsanity.com/paper/2208.05968/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 54 references that are in the map.
- First posted on arXiv 11 Aug 2022.

## Builds on (most cited first, 2 of 2)

- [Minimal Realization Problems for Hidden Markov Models](https://mlsanity.com/paper/1411.3698/) (2014)
- [Model Reduction Methods for Complex Network Systems](https://mlsanity.com/paper/2012.02268/) (2020)

## Abstract (arXiv)

The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones.