Riemann$^2$: Learning Riemannian Submanifolds from Riemannian Data
This addresses the challenge of handling constrained geometric data in latent variable models for domains like robotics and neuroscience, representing a generalization of previous work rather than a foundational breakthrough.
The paper tackled the problem of learning low-dimensional manifolds from geometric data with constraints, such as unit-norm vectors or symmetric positive-definite matrices, by proposing a method to learn Riemannian latent representations that account for data geometry. The result is a model that defines geometry-aware distances and shortest paths in the latent space, ensuring probability mass is only assigned to the data manifold, with applications in robot motion synthesis and brain connectome analysis.
Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches ignore the underlying geometric constraints or fail to provide meaningful metrics in the latent space. To address these limitations, we propose to learn Riemannian latent representations of such geometric data. To do so, we estimate the pullback metric induced by a Wrapped Gaussian Process Latent Variable Model, which explicitly accounts for the data geometry. This enables us to define geometry-aware notions of distance and shortest paths in the latent space, while ensuring that our model only assigns probability mass to the data manifold. This generalizes previous work and allows us to handle complex tasks in various domains, including robot motion synthesis and analysis of brain connectomes.