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This paper investigates different trade-offs between the number of model parameters and enhanced speech qualities by employing several deep tensor-to-vector regression models for speech enhancement. We find that a hybrid architecture, namely CNN-TT, is capable of maintaining a good quality performance with a reduced model parameter size. CNN-TT is composed of several convolutional layers at the bottom for feature extraction to improve speech quality and a tensor-train (TT) output layer on the top to reduce model parameters. We first derive a new upper bound on the generalization power of the convolutional neural network (CNN) based vector-to-vector regression models. Then, we provide experimental evidence on the Edinburgh noisy speech corpus to demonstrate that, in single-channel speech enhancement, CNN outperforms DNN at the expense of a small increment of model sizes. Besides, CNN-TT slightly outperforms the CNN counterpart by utilizing only 32\% of the CNN model parameters. Besides, further performance improvement can be attained if the number of CNN-TT parameters is increased to 44\% of the CNN model size. Finally, our experiments of multi-channel speech enhancement on a simulated noisy WSJ0 corpus demonstrate that our proposed hybrid CNN-TT architecture achieves better results than both DNN and CNN models in terms of better-enhanced speech qualities and smaller parameter sizes.