Riemannian generative decoder

Andreas Bjerregaard*†, Søren Hauberg‡, Anders Krogh†
†University of Copenhagen, ‡Technical University of Denmark
*Correspondance: anje [at] di [dot] ku [dot] dk
Paper Code arXiv 🤗 Dataset

tl;dr: Simpler representation learning on manifolds. We propose a decoder-only framework to learn latents on arbitrary Riemannian manifolds via maximum likelihood and Riemannian optimization. We highlight its use with biological case studies.

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 training a decoder network. By discarding the encoder, we vastly simplify the manifold constraint compared to current approaches which 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.

BibTeX

@inproceedings{bjerregaard2025riemannian,
  title     = {Riemannian generative decoder},
  author    = {Bjerregaard, Andreas and Hauberg, Søren and Krogh, Anders},
  booktitle = {ICML 2025 Workshop on Generative AI and Biology},
  month     = {July},
  year      = {2025}
}