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The recently proposed context-dependent deep neural network hidden Markov models (CD-DNN-HMMs) have been proved highly promising for large vocabulary speech recognition. In this paper, we develop a more advanced type of DNN, which we call the deep tensor neural network (DTNN). The DTNN extends the conventional DNN by replacing one or more of its layers with a double-projection (DP) layer, in which each input vector is projected into two nonlinear subspaces, and a tensor layer, in which two subspace projections interact with each other and jointly predict the next layer in the deep architecture. In addition, we describe an approach to map the tensor layers to the conventional sigmoid layers so that the former can be treated and trained in a similar way to the latter. With this mapping we can consider a DTNN as the DNN augmented with DP layers so that not only the BP learning algorithm of DTNNs can be cleanly derived but also new types of DTNNs can be more easily developed. Evaluation on Switchboard tasks indicates that DTNNs can outperform the already high-performing DNNs with 4-5% and 3% relative word error reduction, respectively, using 30-hr and 309-hr training sets.