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In this paper, we present new methods for parameterizing the connections of neural networks using sums of direct products. We show that low rank parameterizations of weight matrices are a subset of this set, and explore the theoretical and practical benefits of representing weight matrices using sums of Kronecker products. ASR results on a 50 hr subset of the English Broadcast News corpus indicate that the approach is promising. In particular, we show that a factorial network with more than 150doi:10.1109/icassp.2013.6638238 dblp:conf/icassp/FousekRDG13 fatcat:ilhfo73zffgxfecoifbfp5kjxa