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Weak supervision (WS) is a rich set of techniques that produce pseudolabels by aggregating easily obtained but potentially noisy label estimates from various sources. WS is theoretically well-understood for binary classification, where simple approaches enable consistent estimation of pseudolabel noise rates. Using this result, it has been shown that downstream models trained on the pseudolabels have generalization guarantees nearly identical to those trained on clean labels. While this is exciting, users often wish to use WS for\emph {structured prediction}, where the output space consists of more than a binary or multi-class label set: eg rankings, graphs, manifolds, and more. Do the favorable theoretical properties of WS for binary classification lift to this setting? We answer this question in the affirmative for a wide range of scenarios. For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings and tensor decompositions, providing a nearly-consistent noise rate estimator. For labels in constant-curvature Riemannian manifolds, we introduce new invariants that also yield consistent noise rate estimation. In both cases, when using the resulting pseudolabels in concert with a flexible downstream model, we obtain generalization guarantees nearly identical to those for models trained on clean data. Several of our results, which can be viewed as robustness guarantees in structured prediction with noisy labels, may be of independent interest.