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Existing\nuse of LDR matrices in deep learning has applied fixed displacement operators\nencoding forms of shift invariance akin to convolutions. We introduce a class\nof LDR matrices with more general displacement operators, and explicitly learn\nover both the operators and the low-rank component. This class generalizes\nseveral previous constructions while preserving compression and efficient\ncomputation. We prove bounds on the VC dimension of multi-layer neural networks\nwith structured weight matrices and show empirically that our compact\nparameterization can reduce the sample complexity of learning. 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