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Differential privacy is the dominant standard for formal and quantifiable privacy and has been used in major deployments that impact millions of people. Many differentially private algorithms for query release and synthetic data contain steps that reconstruct answers to queries from answers to other queries that have been measured privately. Reconstruction is an important subproblem for such mecha- nisms to economize the privacy budget, minimize error on reconstructed answers, and allow for scalability to high-dimensional datasets. In this paper, we introduce a principled and efficient postprocessing method ReM (Residuals-to-Marginals) for reconstructing answers to marginal queries. Our method builds on recent work on efficient mechanisms for marginal query release, based on making measurements using a residual query basis that admits efficient pseudoinversion, which is an important primitive used in reconstruction. An extension GReM-LNN (Gaussian Residuals-to-Marginals with Local Non-negativity) reconstructs marginals under Gaussian noise satisfying consistency and non-negativity, which often reduces error on reconstructed answers. We demonstrate the utility of ReM and GReM-LNN by applying them to improve existing private query answering mechanisms. 

Abstract Although avian hybrid zones in the Great Plains have been studied for almost 70 years, we know surprisingly little about the fitness costs to hybrids that keep these zones narrow. We compare age ratios in grosbeaks (Pheucticus ludovicianus and P. melanocephalus) and towhees (Pipilo erythropthalums and P. maculatus), two species pairs that differ in their life histories and molt schedules, to evaluate survival between hybrids and parentals. We then contrast molt and migratory divides as possible sources of selection against hybrids. Hybrid grosbeaks had 27%–33% lower survival relative to their parentals, whereas hybrid towhees had survival rates similar to parentals. Age ratio data for hybrid grosbeaks suggest high mortality in older birds, as expected if selection operates after the first year of life. This pattern is consistent with parental species of grosbeaks having contrasting molt schedules relative to migration, suggesting high mortality costs to hybrids driven by molt biology, which are expressed later in life. Contrasts in molt schedules are absent in towhees. While migratory divides may exist for towhees and grosbeaks, the low adult survival of hybrid grosbeaks suggest that molt may be an important and underappreciated source of selection maintaining this and other narrow avian hybrid zones.