Hidden Markov Optimal Transport (HM-OT)
HM-OT is a scalable algorithm for learning cell-state transitions in time-series single-cell datasets using the principle of optimal transport.
Given a time-series of datasets
- A sequence of latent representations
$Q_1, Q_2, \dots, Q_N$ for each timepoint - A set of Markov transition kernels
$\tilde{T}^{(i,i+1)}$ between adjacent timepoints
For single-cell transcriptomics, this corresponds to learning:
- A set of latent cell states at each time
- The least-action transition maps that explain cellular differentiation across time
(Figure 1: HM-OT infers cell-state transitions between timepoints using optimal transport with a Hidden Markov structure.)
To get started, clone the repository and install dependencies with:
git clone https://github.com/raphael-group/HM-OT.git
cd HM-OT
# Install dependencies
pip install -r requirements.txtThe main method for running HM-OT is in src/HiddenMarkovOT.py. See the demo notebook in notebooks/ for a full example.
Folder Structure
HM-OT/
├── src/ # Source directory
│ ├── FRLC/ # Low-rank OT solver
│ ├── utils/ # Utility functions
└── clustering.py # Functions for computing clusterings
└── util_LR.py # Utilities for HM-OT preprocessing
└── util_zf.py # Other utilities
│ └── HiddenMarkovOT.py # Main HM-OT interface
└── plotting.py # Plotting functions
├── notebooks/ # Example notebooks
└── HM_OT_SingleCell_Demo.ipynb # Demo for diff-maps on single-cell
├── images/ # Visual figures
│ └── Figure1.pdf
├── requirements.txt
└── README.mdContact
If you have any questions or difficulties at all, feel free to reach out to Peter Halmos (ph3641@princeton.edu) or Julian Gold (jg7090@princeton.edu). We're happy to help!
If this work has been helpful to your research, feel free to cite our preprint:
@article{Halmos2025,
title = {Learning Latent Trajectories in Developmental Time Series with Hidden-Markov Optimal Transport},
url = {http://dx.doi.org/10.1101/2025.02.14.638351},
DOI = {10.1101/2025.02.14.638351},
publisher = {Cold Spring Harbor Laboratory},
author = {Halmos, Peter and Gold, Julian and Liu, Xinhao and Raphael, Benjamin J.},
year = {2025},
month = feb
}