Geodesic Calculus on Latent Spaces
This work addresses the challenge of geometric analysis in machine learning for researchers and practitioners using latent representations, though it is incremental as it builds on existing implicit representation methods.
The paper tackles the problem of performing geometric computations on latent manifolds of autoencoders by proposing a framework for discrete Riemannian calculus, enabling robust computation of geodesic paths and exponential maps. It evaluates the approach on synthetic and real data, demonstrating its applicability across different autoencoders and Riemannian geometries.
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.