Hyperbolic Curvature as an Inductive Bias for Latent Space Flow Matching

Learning image representations that respect the intrinsic geometry of data is crucial for capturing hierarchical semantic structure, yet generative transport is typically performed in Euclidean spaces where this structure is not preserved. In this work, we propose a geometry-aware generative framework that combines hyperbolic representation learning with Riemannian Flow Matching to perform generative transport directly in hyperbolic latent space. Instead of learning generative dynamics in pixel space or Euclidean latents, we transport samples directly on the manifold produced by a pretrained hyperbolic autoencoder, preserving geometric organization and yielding more stable samples and improved FID score, compared to Euclidean latent transport. We further investigate curvature as a controllable geometric inductive bias and observe a trade-off between generation realism and diversity, where moderate curvature yields more coherent samples, and larger curvature allows visual variation at the cost of stability, highlighting how latent geometry shapes generative transport.