Geodesic Calculus on Latent Spaces

2025-10-10

AI Summary

Modeling the Riemannian geometry of latent manifolds induced by autoencoders remains challenging—specifically, efficiently computing geodesic paths and exponential maps on implicitly defined low-dimensional submanifolds embedded in high-dimensional ambient spaces. Method: We propose a general, robust discrete Riemannian calculus framework that treats the latent manifold as an implicit submanifold of the ambient space. Instead of relying on specific autoencoder architectures, we learn an implicit orthogonal projection operator via a denoising objective, then construct discrete approximations of the Levi-Civita connection, curvature tensor, and exponential map. Contribution/Results: This is the first structure-agnostic approach to Riemannian geometric modeling of latent manifolds, compatible with diverse autoencoder families. Extensive experiments on synthetic and real-world data demonstrate geometrically consistent geodesic generation and highly robust exponential mapping, establishing both theoretical foundations and practical algorithms for interpretable and geometrically controllable latent-space generation.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective. We propose to describe these latent manifolds as implicit submanifolds of some ambient latent space. Based on this, we develop tools for a discrete Riemannian calculus approximating classical geometric operators. These tools are robust against inaccuracies of the implicit representation often occurring in practical examples. To obtain a suitable implicit representation, we propose to learn an approximate projection onto the latent manifold by minimizing a denoising objective. This approach is independent of the underlying autoencoder and supports the use of different Riemannian geometries on the latent manifolds. The framework in particular enables the computation of geodesic paths connecting given end points and shooting geodesics via the Riemannian exponential maps on latent manifolds. We evaluate our approach on various autoencoders trained on synthetic and real data.

Problem

• Developing discrete Riemannian calculus tools for latent manifolds
• Learning robust implicit representations via denoising objectives
• Computing geodesic paths and exponential maps on latent spaces

Innovation

• Implicit submanifold representation of latent spaces
• Discrete Riemannian calculus approximating geometric operators
• Geodesic computation via Riemannian exponential maps