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The goal of this paper is to find sparse and representative spatial priors that can be applied to part-based object localization. Assuming a GMRF prior over part configurations, we construct the graph structure of the prior by regressing the position of each part on all other parts, and selecting the neighboring edges using a Lasso-based method. This approach produces a prior structure which is not only sparse, but also faithful to the spatial dependencies that are observed in training data. We evaluate the representation power of the learned prior structure in two ways: first is drawing samples from the prior, and comparing them with the samples produced by the GMRF priors of other structures; second is comparing the results when applying different priors to a facial components localization task. We show that the learned graph captures meaningful geometrical variations with significantly sparser structure and leads to better parts localization results.