Title: PolyCNN: Learning Seed Convolutional Filters. Abstract: In this work, we propose the polynomial convolutional neural network (PolyCNN), as a new design of a weight-learning efficient variant of the traditional CNN. The biggest advantage of the PolyCNN is that at each convolutional layer, only one convolutional filter is needed for learning the weights, which we call the seed filter, and all the other convolutional filters are the polynomial transformations of the seed filter, which is termed as an early fan-out. Alternatively, we can also perform late fan-out on the seed filter response to create the number of response maps needed to be input into the next layer. Both early and late fan-out allow the PolyCNN to learn only one convolutional filter at each layer, which can dramatically reduce the model complexity by saving 10x to 50x parameters during learning. While being efficient during both training and testing, the PolyCNN does not suffer performance due to the non-linear polynomial expansion which translates to richer representational power within the convolutional layers. By allowing direct control over model complexity, PolyCNN provides a flexible trade-off between performance and efficiency. We have verified the on-par performance between the proposed PolyCNN and the standard CNN on several visual datasets, such as MNIST, CIFAR-10, SVHN, and ImageNet.