Riemannian generative decoder

2025-06-23

AI Summary

Existing Riemannian representation learning methods suffer from manifold specificity, optimization challenges, and geometric constraints imposed by encoder-based architectures, hindering simultaneous generality and interpretability. This paper proposes an encoder-free Riemannian generative decoding framework: a decoder network is jointly trained via Riemannian optimization to directly maximize the likelihood of latent variables on the target manifold, yielding a geometrically aligned and interpretable latent space. By eliminating reliance on manifold-specific designs and explicit geometric regularizers, our approach is architecture-agnostic—compatible with standard neural backbones—and supports modeling diverse non-Euclidean structures, including synthetic diffusion processes, mitochondrial DNA migration, and cell-cycle dynamics. Evaluated on three real-world scenarios, the learned representations accurately capture intrinsic non-Euclidean geometry, exhibit improved training stability, and deliver significantly enhanced interpretability.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

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.

Problem

• Avoids approximating densities on Riemannian manifolds
• Simplifies manifold constraints by discarding encoder
• Captures intrinsic non-Euclidean structure in data

Innovation

• Riemannian generative decoder for manifold-valued latents
• Discards encoder to simplify manifold constraints
• Compatible with existing architectures and interpretable