  The choice of approximate posterior distributions plays a central role in
stochastic variational inference (SVI). One effective solution is the use of
normalizing flows \cut{defined on Euclidean spaces} to construct flexible
posterior distributions. However, one key limitation of existing normalizing
flows is that they are restricted to the Euclidean space and are ill-equipped
to model data with an underlying hierarchical structure. To address this
fundamental limitation, we present the first extension of normalizing flows to
hyperbolic spaces. We first elevate normalizing flows to hyperbolic spaces
using coupling transforms defined on the tangent bundle, termed Tangent
Coupling ($\mathcal{TC}$). We further introduce Wrapped Hyperboloid Coupling
($\mathcal{W}\mathbb{H}C$), a fully invertible and learnable transformation
that explicitly utilizes the geometric structure of hyperbolic spaces, allowing
for expressive posteriors while being efficient to sample from. We demonstrate
the efficacy of our novel normalizing flow over hyperbolic VAEs and Euclidean
normalizing flows. Our approach achieves improved performance on density
estimation, as well as reconstruction of real-world graph data, which exhibit a
hierarchical structure. Finally, we show that our approach can be used to power
a generative model over hierarchical data using hyperbolic latent variables.
