Riemannian generative decoder:
simpler representation learning on manifolds

Riemannian generative decoder
Andreas Bjerregaard, Søren Hauberg and Anders Krogh
Transactions on Machine Learning Research, 2026
J2C Certification; planned presentation for ICML 2026, July 6-11
https://openreview.net/forum?id=vuPMXg1FDT

Abstract: Riemannian representation learning typically relies on approximating densities on chosen manifolds. This involves optimizing difficult objectives, potentially harming models. To completely circumvent this issue, we introduce the Riemannian generative decoder which finds manifold-valued maximum likelihood latents with a Riemannian optimizer while jointly training a decoder network. By discarding the encoder, we vastly simplify the manifold constraint compared to current approaches which can often only handle few specific manifolds. We validate our approach on three case studies — a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells undergoing a cell division cycle — each showing that learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure. Our method requires only a decoder, is compatible with existing architectures, and yields interpretable latent spaces aligned with data geometry.

Running the project

To reproduce results and figures from the paper, follow these steps. All notebooks will automatically fetch required data.

1. Setup environment

   git clone --recurse-submodules https://github.com/fvk220/riemannian-generative-decoder.git
   pip install -e data/geoopt

2. Cell cycle in human fibroblasts [1]

   • Notebook: 1) CellCycle.ipynb

3. Synthetic branching diffusion process [2]

   • Notebook: 2) BranchingDiffusion.ipynb

4. Human mitochondrial DNA [3]

   • Notebooks: 3) hmtDNA*.ipynb
   • For updating mitochondrial sequences, refer to Scraping new mitochondrial sequences.

All the used data can also be downloaded directly from 🤗 HuggingFace without cloning the repository.

Scraping new mitochondrial sequences

If you wish to make an updated scrape of human mitochondrial sequences, run the following Python script:

from Bio import Entrez
from tqdm import tqdm

Entrez.email = "your_email@gmail.com"
query = '(00000015400[SLEN] : 00000016700[SLEN]) AND "Homo"[Organism] AND mitochondrion[FILT] NOT (unverified[All Fields] OR ("Homo sapiens subsp. \'Denisova\'"[Organism] OR Homo sp. Altai[All Fields]) OR ("Homo sapiens subsp. \'Denisova\'"[Organism] OR Denisova hominin[All Fields]) OR neanderthalensis[All Fields] OR heidelbergensis[All Fields] OR consensus[All Fields])'

handle = Entrez.esearch(db="nuccore", term=query, retmax=100000)
record = Entrez.read(handle)
id_list = record["IdList"]
with open("seqs.fasta", "w") as outfile:
    for start in tqdm(range(0, len(id_list), 500), desc="Downloading sequences"):
        fetch_ids = id_list[start:start+500]
        handle = Entrez.efetch(db="nuccore", id=fetch_ids, rettype="fasta", retmode="text")
        records = handle.read()
        outfile.write(records)

Then, use haplogrep3 [4] to annotate the sequences using either an RSRS or rCRS tree:

$ haplogrep3 classify --in "seqs.fasta" --out seqs_rsrs.txt --tree phylotree-rsrs@17.0 --extend-report
$ haplogrep3 classify --in "seqs.fasta" --out seqs_rcrs.txt --tree phylotree-fu-rcrs@1.2 --extend-report

This generates a file with mutations compared to a reference sequence, as well as more metadata — see the haplogrep3 documentation for details. Additional metadata (e.g., geographic location) can be added during a second GenBank scrape; merging such data with the haplogrep3 output produces the zipped files data/hmtDNA/63k_out_rcrs.txt.gz and data/hmtDNA/63k_out_rsrs.txt.gz. Upon approval, this pipeline will be made available as a standalone script, and a resulting snapshot will be published via Zenodo.

Requirements

The code has been tested with Python 3.11.11 using the packages from requirements.txt. Use the geoopt 0.5.1 submodule (including a sphere origin function) with $ pip install -e data/geoopt.

Acknowledgements

We thank the developers of geoopt [5] and haplogrep3 [4], the QuickGO [6] and MITOMAP projects [3], as well as Riba et al. [1] and Mathieu et al. [2] for making their software and data publicly available.

We further thank Viktoria Schuster, Iñigo Prada-Luengo, Yan Li, Adrián Sousa-Poza and Valentina Sora for helpful discussions. This work was funded by the Novo Nordisk Foundation (NNF) through the Center for Basic Machine Learning in Life Science (NNF grant NNF20OC0062606) and the Pioneer Centre for AI (DNRF grant number P1). A.K. was further supported by NNF grants NNF20OC0059939 and NNF20OC0063268.

Citation

@article{
bjerregaard2026riemannian,
title={Riemannian Generative Decoder},
author={Andreas Bjerregaard and S{\o}ren Hauberg and Anders Krogh},
journal={Transactions on Machine Learning Research},
issn={2835-8856},
year={2026},
url={https://openreview.net/forum?id=vuPMXg1FDT},
note={J2C Certification}
}

Bibliography

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2. E. Mathieu, C. Le Lan, C. J. Maddison, R. Tomioka, and Y. W. Teh. "Continuous hierarchical representations with Poincaré variational auto-encoders". Advances in Neural Information Processing Systems, 32, 2019.

3. MITOMAP: A Human Mitochondrial Genome Database. http://www.mitomap.org, 2023. Database.

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