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Currently, how to deal with the massive garbage produced by various human activities is a hot topic all around the world. In this paper, a preliminary and essential step is to classify the garbage into different categories. However, the mainstream waste classification mode relies heavily on manual work, which consumes a lot of labor and is very inefficient. With the rapid development of deep learning, convolutional neural networks (CNN) have been successfully applied to various application fields. Therefore, some researchers have directly adopted CNNs to classify garbage through their images. However, compared with other images, the garbage images have their own characteristics (such as inter-class similarity, intra-class variance and complex background). Thus, neglecting these characteristics would impair the classification accuracy of CNN. To overcome the limitations of existing garbage image classification methods, a Depth-wise Separable Convolution Attention Module (DSCAM) is proposed in this paper. In DSCAM, the inherent relationships of channels and spatial positions in garbage image features are captured by two attention modules with depth-wise separable convolutions, so that our method could only focus on important information and ignore the interference. Moreover, we also adopt a residual network as the backbone of DSCAM to enhance its discriminative ability. We conduct the experiments on five garbage datasets. The experimental results demonstrate that the proposed method could effectively classify the garbage images and that it outperforms some classical methods.