A novel deep architecture, the tensor deep stacking network (T-DSN), is presented. The T-DSN consists of multiple, stacked blocks, where each block contains a bilinear mapping from two hidden layers to the output layer, using a weight tensor to incorporate higher order statistics of the hidden binary (([0,1])) features. A learning algorithm for the T-DSN's weight matrices and tensors is developed and described in which the main parameter estimation burden is shifted to a convex subproblem with a closed-form solution. Using an efficient and scalable parallel implementation for CPU clusters, we train sets of T-DSNs in three popular tasks in increasing order of the data size: handwritten digit recognition using MNIST (60k), isolated state/phone classification and continuous phone recognition using TIMIT (1.1 m), and isolated phone classification using WSJ0 (5.2 m). Experimental results in all three tasks demonstrate the effectiveness of the T-DSN and the associated learning methods in a consistent manner. In particular, a sufficient depth of the T-DSN, a symmetry in the two hidden layers structure in each T-DSN block, our model parameter learning algorithm, and a softmax layer on top of T-DSN are shown to have all contributed to the low error rates observed in the experiments for all three tasks.