On the Curvature of Deep Generative Models
According to the manifold hypothesis, high-dimensional data, despite its apparent complexity, often adheres to a simpler underlying structure, typically represented as a manifold, or a lower-dimensional surface embedded within the larger dimensional space. Performing computations within these high-dimensional environments presents significant challenges. A practical approach is to parameterize the surface in $\mathcal{X}$ by a low-dimensional variable $\mathbf{z} \in \mathcal{Z}$ created with a suitable smooth generator function $f : \mathcal{Z} \to \mathcal{X}$.
