# Geometric Learning of Hidden Markov Models via a Method of Moments Algorithm

arXiv: https://arxiv.org/abs/2207.00818 · HTML: https://mlsanity.com/paper/2207.00818/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 33 references that are in the map.
- First posted on arXiv 2 Jul 2022.

## Builds on (most cited first, 11 of 11)

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [geomstats: a Python Package for Riemannian Geometry in Machine Learning](https://mlsanity.com/paper/1805.08308/) (2018)
- [Partially Observed Markov Decision Processes. Problem Sets and Internet Supplement](https://mlsanity.com/paper/1604.08127/) (2016)
- [On learning parametric-output HMMs](https://mlsanity.com/paper/1302.6009/) (2013)
- [Learning Hidden Markov Models from Pairwise Co-occurrences with Application to Topic Modeling](https://mlsanity.com/paper/1802.06894/) (2018)
- [Geometric Matrix Midranges](https://mlsanity.com/paper/1907.04188/) (2019)
- [Inductive Geometric Matrix Midranges](https://mlsanity.com/paper/2006.01508/) (2020)
- [Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels](https://mlsanity.com/paper/2011.13680/) (2020)
- [Online learning of Riemannian hidden Markov models in homogeneous Hadamard spaces](https://mlsanity.com/paper/2102.07771/) (2021)
- [Asymptotically Efficient Identification of Known-Sensor Hidden Markov Models](https://mlsanity.com/paper/1702.00155/) (2017)

## Built on by (most cited first, 2 of 2)

- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)
- [Differential geometry with extreme eigenvalues in the positive semidefinite cone](https://mlsanity.com/paper/2304.07347/) (2023)

## Abstract (arXiv)

We present a novel algorithm for learning the parameters of hidden Markov models (HMMs) in a geometric setting where the observations take values in Riemannian manifolds. In particular, we elevate a recent second-order method of moments algorithm that incorporates non-consecutive correlations to a more general setting where observations take place in a Riemannian symmetric space of non-positive curvature and the observation likelihoods are Riemannian Gaussians. The resulting algorithm decouples into a Riemannian Gaussian mixture model estimation algorithm followed by a sequence of convex optimization procedures. We demonstrate through examples that the learner can result in significantly improved speed and numerical accuracy compared to existing learners.