The paper introduces parsimoniously activated dictionary learning (PADL), a method imposing global regularization on the number of activated dictionary atoms. It demonstrates that PADL is equivalent to maximum a posteriori estimation under a structured generative model with auxiliary latent variables. This equivalence enables the derivation of generalization guarantees that are difficult to obtain from the original formulation. The authors provide an analytical characterization of the tradeoff between sparsity, storage cost, and reconstruction accuracy. This framework allows for data-driven estimation of optimal hyperparameters without manual tuning. An efficient and interpretable PADL algorithm is developed based on this theoretical connection. Experimental results show improved reconstruction performance under comparable sparsity levels on visual benchmarks. The method also demonstrates practical utility in accelerating inference for vision-language models.