Max-Affine Spline Insights into Deep Generative Networks
This provides theoretical insights into deep generative models for researchers, though it is incremental as it builds on existing spline theory.
The authors analyzed Generative Deep Networks (GDNs) through a max-affine spline framework to characterize their manifold properties, approximation errors, and disentanglement conditions, while deriving formulas for output probability densities and likelihood computation that reveal limitations in modeling low-entropy/multimodal distributions.
We connect a large class of Generative Deep Networks (GDNs) with spline operators in order to derive their properties, limitations, and new opportunities. By characterizing the latent space partition, dimension and angularity of the generated manifold, we relate the manifold dimension and approximation error to the sample size. The manifold-per-region affine subspace defines a local coordinate basis; we provide necessary and sufficient conditions relating those basis vectors with disentanglement. We also derive the output probability density mapped onto the generated manifold in terms of the latent space density, which enables the computation of key statistics such as its Shannon entropy. This finding also enables the computation of the GDN likelihood, which provides a new mechanism for model comparison as well as providing a quality measure for (generated) samples under the learned distribution. We demonstrate how low entropy and/or multimodal distributions are not naturally modeled by DGNs and are a cause of training instabilities.