Dictionary-Transform Generative Adversarial Networks
Angshul Majumdar
TL;DR
DT-GAN replaces neural generators and discriminators with a sparse dictionary $D$ and an energy-based transform $T$, yielding a fully model-based adversarial framework that is mathematically well posed. The authors prove existence of Nash equilibria, identifiability of the dictionary up to permutation and sign, and finite-sample stability, while demonstrating robust recovery of sparse structure on mixture data under identical training budgets to a neural GAN. By grounding adversarial learning in sparse modeling, the work achieves interpretable, stable, and provably correct behavior for data distributions with sparse synthesis structure, and suggests extensions to robust and hierarchical dictionary-transform architectures. The scope is deliberately focused on sparse-synthesis regimes rather than universal nonlinear manifolds, offering a principled alternative with strong theoretical guarantees and practical stability in this setting.
Abstract
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.