## 7 Conclusions  
We proposed a training method which alternatively composes and decomposes the filters in the training stage for better training of low rank filters. As an exemplary case, we showed that a rank-1 CNN can be trained with the proposed method, which cannot be trained with conventional training methods. We used 3-D rank-1 filters in convolutional neural networks in the training phase, so that the redundancy in the filters are reduced by the rank deficient property of the rank-1 filters. The proposed training method updates the 3-D filter parameters and projects them back onto the rank-1 subspace at each epoch to find good parameter values for the 1-D vectors which constitute the 3-D filters. At test time, the trained 1-D vectors can be used directly as 1-D filters which filter the image by 1-D convolution instead of the 3-D convolution for fast inference. We showed in the experiments that the accuracy performance of the rank-1 CNN is almost the same as the standard CNN while reducing the number of parameters up to about 11%, and the number of operations up to about 12% in the convolution filters compared with the standard CNN. We also showed that the proposed method can also be used in the ResNet structure and  
Figure 11: Comparison of mean validation accuracies on the CIFAR10 dataset with the proposed and the standard training methods for the MobileNet. (**a**) Mean validation accuracies of the proposed training method (vertical bars show the positive standard deviation); (**b**) mean validation accuracies of the standard training method (vertical bars show the positive standard deviation).  
showed that the proposed training method can be utilized for a better training of the MobileNet. This suggests the possibility that the proposed rank-1 training method can also be used with diverse structures such as the ResNet or other small-sized networks, to obtain a more efficient network structure.  
In this paper, we applied the proposed training method to well-known deep learning models that can be factorized. Whether the proposed training method can be applied to other complicated models still leaves much room. Combining the proposed method with the existing training method and applying it to other complex deep learning models can be an additional topic of this study. Moreover, the experimental results showed that the inference time was not reduced as much as the decrease in the number of parameters, which is due to the fact that existing deep learning frameworks are not optimized for factorized models. Therefore, in order to accelerate the inference speed, further studies on hardware designs that can effectively adopt factorized models should be carried out in the future.  
conceptualization, S.L., H.K. and J.Y.; methodology, S.L., B.J.; software, S.L., H.K.; validation, B.J., J.Y.; formal analysis, H.K., J.Y.; investigation, J.Y.; resources, S.L., B.J.; writing--original draft preparation, S.L., H.K.; writing--review and editing, S.L., B.S. J.Y.; visualization, S.L.; supervision, J.Y.; project administration, J.Y. All authors have read and agreed to the published version of the manuscript.  
The work of [PERSON] was supported in part by the National Research Foundation of Korea under Grant NRF-2015R1A5A1009350 and the work of [PERSON] was supported by the Basic Science Research Program through the National Research Foundation of Korea under Grant NRF-2019R1 II1A3A01060150.  
Data sharing not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.