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We propose a weakly-supervised framework for the semantic segmentation of circular-scan synthetic-aperture-sonar (CSAS) imagery. The first part of our framework is trained in a supervised manner, on image-level labels, to uncover a set of semi-sparse, spatially-discriminative regions in each image. The classification uncertainty of each region is then evaluated. Those areas with the lowest uncertainties are then chosen to be weakly labeled segmentation seeds, at the pixel level, for the second part of the framework. Each of the seed extents are progressively resized according to an unsupervised, information-theoretic loss with structured-prediction regularizers. This reshaping process uses multi-scale, adaptively-weighted features to delineate class-specific transitions in local image content. Content-addressable memories are inserted at various parts of our framework so that it can leverage features from previously seen images to improve segmentation performance for related images. We evaluate our weakly-supervised framework using real-world CSAS imagery that contains over ten seafloor classes and ten target classes. We show that our framework performs comparably to nine fully-supervised deep networks. Our framework also outperforms eleven of the best weakly-supervised deep networks. We achieve state-of-the-art performance when pre-training on natural imagery. The average absolute performance gap to the next-best weakly-supervised network is well over ten percent for both natural imagery and sonar imagery. This gap is found to be statistically significant., Comment: Submitted to the IEEE Journal of Oceanic Engineering