## 1 Introduction  
Deep learning-based computer vision shows good performance in various computer vision areas such as image segmentation [1, 2], image synthesis [3], facial recognition [4], classification [5], person re-identification [6], and object detection [7, 8]. However, in spite of the remarkable achievements in difficult computer vision tasks, conventional deep convolutional neural networks (CNNs) use a high number of parameters which limits their use on devices with limited resources such as smartphones, embedded systems, etc. Even though it has been known that there exist a lot of redundancy between the parameters and the feature maps in deep models, over-parametrized CNN models are used due to the reason that over-parametrization makes the training of the network easier as has been shown in the experiments in [9]. The reason for this phenomenon is believed to be due to a better gradient flow in over-parametrized networks. Meanwhile, it has been shown in [10] that even with the use of regularization methods, there still exists excessive capacity in trained networks, which again implies the fact that the redundancy between the parameters is still large. Therefore, many researches focus on finding a better network structure so that the parameters can be expressed in a structured subspace with smaller number of coefficients. The research topic on compressing large-scale deep learning models is increasing in importance as it is necessary to use compressed deep learning models in edge devices such as smartphones and IoT devices. While early works focused on compressing the parameters of pre-trained large scale deep learning models [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22], studies are also actively under way to limit the number of parameters by proposing small networks in the first place [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37]. Most recently, researches have prevailed on how to efficiently use these compressed models on edge devices [38; 39; 40; 41]. We will provide a detailed overview of these research trends in Section 2. In this paper, we propose a training method for the training of low-rank convolutional neural networks, which we call the alternating tensor compose-decompose (ATCD) method. The proposed training method can better train compressed low rank models than existing training methods, thus obtaining a compressed deep learning model with higher performance. In general, when training deep learning models, the same structure of the neural network is used during the training and the testing stages. Conventional tensor decomposing networks are trained with the fixed-structure based training method, i.e., they are trained in the decomposed form. We call the conventional training method the fixed-structure based training method. In comparison, the proposed training method do not use a fixed structure of neural network in the training stage, but allows the tensors to be alternatingly composed and decomposed so that a better gradient flow can flow through the tensors in the backpropagation step. This better gradient flow results in better parameter values than with conventional training method so that the compressed model can achieve a higher performance. As an example of the proposed training method, we apply it to the rank-1 CNN, where the rank-1 CNN is iteratively and alternatingly composed into a 3-D rank-1 CNN structure and decomposed into 1-D vectors in the training stage, where the 3-D rank-1 filters are constructed by the outer products of the 1-D vectors. The number of parameters in the 3-D rank-1 filters are the same as in the 3-D filters in standard CNNs, allowing a good gradient flow in the backpropagation stage. The difference with the backpropagation stage in standard CNNs is that the gradient flow flows also through the 1-D vectors from which the 3-D rank-1 filters are constructed, updating the parameters in the 1-D vectors also. After the backpropagation step, the 3-D filters lose their rank-1 property.