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In this paper, we consider the estimation of hyperparameters for regularization terms commonly used for obtaining structured sparse parameters in signal estimation problems, such as signal denoising. By considering the convex regularization terms as negative log-densities, we propose approximate maximum likelihood estimation for estimating parameters for continuous log-supermodular distributions, which is a key property that many sparse priors have. We then show how "perturb-and-MAP" ideas based on the Gumbel distribution and efficient discretization can be used to approximate the log-partition function for these models, which is a crucial step for approximate maximum likelihood estimation. We illustrate our estimation procedure on a set of experiments with flow-based priors and signal denoising.