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Deep neural networks have achieved outstanding performance in computer vision tasks. Convolutional Neural Networks (CNNs) typically operate in the spatial domain with raw images, but in practice, images are usually stored and transmitted in their compressed representation where JPEG is one of the most widely used encoder. Also, these networks are computationally intensive and slow. This paper proposes performing the learning and inference processes in the compressed domain in order to reduce the computational complexity and improve the speed of popular CNNs. For this purpose, a novel graph-based frequency channel selection method is proposed to identify and select the most important frequency channels. The computational complexity is reduced by retaining the important frequency components and discarding the insignificant ones as well as eliminating the unnecessary layers of the network. Experimental results show that the modified ResNet-50 operating in the compressed domain is up to 70% faster than the spatial-based traditional ResNet-50 while resulting in similar classification accuracy. Moreover, this paper proposes a preprocessing step with partial encoding to improve the resilience to distortions caused by low-quality encoded images. Finally, we show that training a network with highly compressed data can achieve a good classification accuracy with up to 93% reduction in the storage requirements of the training data. [ABSTRACT FROM AUTHOR]