Dictionary-Transform Generative Adversarial Networks
This provides a principled, interpretable alternative for sparse data distributions, addressing instability issues in adversarial learning, though it is incremental as it builds on sparse modeling rather than replacing neural GANs universally.
The authors tackled the theoretical fragility and instability of classical GANs by introducing Dictionary-Transform GANs (DT-GAN), a model-based adversarial framework with linear operators, and showed that it admits Nash equilibria, provably recovers underlying structure, and exhibits stable training where standard GANs degrade.
Generative adversarial networks (GANs) are widely used for distribution learning, yet their classical formulations remain theoretically fragile, with ill-posed objectives, unstable training dynamics, and limited interpretability. In this work, we introduce \emph{Dictionary-Transform Generative Adversarial Networks} (DT-GAN), a fully model-based adversarial framework in which the generator is a sparse synthesis dictionary and the discriminator is an analysis transform acting as an energy model. By restricting both players to linear operators with explicit constraints, DT-GAN departs fundamentally from neural GAN architectures and admits rigorous theoretical analysis. We show that the DT-GAN adversarial game is well posed and admits at least one Nash equilibrium. Under a sparse generative model, equilibrium solutions are provably identifiable up to standard permutation and sign ambiguities and exhibit a precise geometric alignment between synthesis and analysis operators. We further establish finite-sample stability and consistency of empirical equilibria, demonstrating that DT-GAN training converges reliably under standard sampling assumptions and remains robust in heavy-tailed regimes. Experiments on mixture-structured synthetic data validate the theoretical predictions, showing that DT-GAN consistently recovers underlying structure and exhibits stable behavior under identical optimization budgets where a standard GAN degrades. DT-GAN is not proposed as a universal replacement for neural GANs, but as a principled adversarial alternative for data distributions that admit sparse synthesis structure. The results demonstrate that adversarial learning can be made interpretable, stable, and provably correct when grounded in classical sparse modeling.