Conditionally Strongly Log-Concave Generative Models
Florentin Guth Etienne Lempereur Joan Bruna Stéphane Mallat
There is a growing gap between the impressive results of deep image
generative models and classical algorithms that offer theoretical guarantees.
The former suffer from mode collapse or memorization issues, limiting their
application to scientific data. The latter require restrictive assumptions such
as log-concavity to escape the curse of dimensionality. We partially bridge
this gap by introducing conditionally strongly log-concave (CSLC) models, which
factorize the data distribution into a product of conditional probability
distributions that are strongly log-concave. This factorization is obtained
with orthogonal projectors adapted to the data distribution. It leads to
efficient parameter estimation and sampling algorithms, with theoretical
guarantees, although the data distribution is not globally log-concave. We show
that several challenging multiscale processes are conditionally log-concave
using wavelet packet orthogonal projectors. Numerical results are shown for
physical fields such as the $\varphi^4$ model and weak lensing convergence maps
with higher resolution than in previous works.