Most variational formulations for structure-texture image decomposition force the structure images to have small norm in some functional spaces and to share a common notion of edges, i.e., large-gradients or large-intensity differences. However, such a definition makes it difficult to distinguish structure edges from oscillations that have fine spatial scale but high contrast. In this paper, we introduce a new model by learning deep variational priors for structure images without explicit training data. An alternating direction method of a multiplier algorithm and its modular structure are adopted to plug deep variational priors into an iterative smoothing process.