Jungho Yoon

Costanza Conti, Lucia Romani, Jungho Yoon

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: i) reproduction of $N$ exponential polynomials implies approximate sum rules of order $N$; ii) generation of $N$ exponential polynomials implies approximate sum rules of order $N$, under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; iii) reproduction of an $N$-dimensional space of exponential polynomials and asymptotical similarity imply approximation order $N$; iv) the sequence of basic limit functions of a non-stationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme.

Hyein Kim, Jungho Yoon, Byeongseon Jeong et al.

In this paper, we propose a convolutional neural network(CNN) with 3-D rank-1 filters which are composed by the outer product of 1-D filters. After being trained, the 3-D rank-1 filters can be decomposed into 1-D filters in the test time for fast inference. The reason that we train 3-D rank-1 filters in the training stage instead of consecutive 1-D filters is that a better gradient flow can be obtained with this setting, which makes the training possible even in the case where the network with consecutive 1-D filters cannot be trained. The 3-D rank-1 filters are updated by both the gradient flow and the outer product of the 1-D filters in every epoch, where the gradient flow tries to obtain a solution which minimizes the loss function, while the outer product operation tries to make the parameters of the filter to live on a rank-1 sub-space. Furthermore, we show that the convolution with the rank-1 filters results in low rank outputs, constraining the final output of the CNN also to live on a low dimensional subspace.