Deep neural networks (DNNs) have been widely used for data processing and analysis nowadays. Many computational techniques have been proposed to improve the energy efficiency of executing DNNs, which is critical for emerging smart edge applications. This article presents a speculative computation approach to improving the energy efficiency of DNN computations. The proposed approach employs the techniques of input channel partitioning and threshold-based negative masking to predict and eliminate unnecessary computations. Moreover, a systematic procedure of threshold optimization is proposed to achieve the best tradeoff between the energy and accuracy performance. Finally, an energy-efficient DNN processor architecture was designed and implemented to support the proposed speculative computation approach. The experimental results show that the proposed DNN processor with speculative computation can enhance the energy efficiency of the processor by 22.8%, with only 0.96% accuracy degradation and 1% implementation overhead.