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Matrix factorization methods are widely used for extracting latent factors for low rank matrix completion and rating prediction problems arising in recommender systems of on-line retailers. Most of the exist- ing models are based on L2 delity (quadratic functions of factorization error). In this work, a coordinate descent (CD) method is developed for matrix factorization under L1 delity so that the related minimization is done one variable at a time and the factorization error is sparsely distributed. In low rank random matrix completion and rating predic- tion of MovieLens 100k datasets, the CDL1 method shows remarkable stability and accuracy under gross corruption of training (observation) data while the L2 delity based methods rapidly deteriorate. A closed form analytical solution is found for the one-dimensional L1-delity sub- problem, and is used as a building block of CDL1 algorithm whose con- vergence is analyzed. A connection with robust principal component analysis is drawn.

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In the early 1980s of twenty century, Professor Zhang Zhongfu, an expert in graph theory, raised a question in his research: how many points of intersection of diagonal lines are there inside a regular N polygon, hoping to find out a formula of count. It has been more than 20 years since the question was raised, which has aroused the interest of quite a lot of experts, scholars and those who love mathematics, but still remains unsolved. When N is an odd number, proposition 1 can be derived from formula of counting based on proposition 2, which can also be verified by the programme the author has made. Proposition 1: when N is an odd number, there is no concurrence of 3 or more than 3 diagonal lines of regular N polygon. Proposition 2: when N is an odd number, the number of intersection points of inner diagonal lines of regular N polygon is: &a_n=c^{4}_{n}=frac{1}{24}(n-1)(n-2)(n-3)$ But when N is an even number, it becomes quite complicated. When N are some special even numbers referred to by Chang Jianming, there are some laws: the research is made from the perspective of prime factorization of figures. The writer primarily studies the issue by making use of geometrical drawing and imitation by computer programme, so that the programme is improved and the imitating solutions of the numbers of intersection points of inner diagonal lines of regular N polygon with specific programmes can be solved. The solution is popularizable. The writer has worked out the numbers of intersection points of inner diagonal lines of regular 2N polygon when N 50, if conditions of computers in schools allow. By comparing the counting between drawing and computer imitation, the writer has also found out some interesting laws about the numbers of intersection points of inner diagonal lines of regular N polygon, and has put forward some hypotheses.

This work focuses on the problem of high quality style transfer for Chinese characters, converting Chinese printed with standard typograph into those in calligraphic style. Traditional approaches include employing derivative network structures of CNN, RNN, and GAN, yet generation results are not satisfactory enough. In this paper, we proposed an integrated system of pix2pix structure, auxiliary classifier, and WGAN. Trials were run to evaluate the performance of traditional CNN and GAN. Cross-comparative sets of experiments among GAN, AC-GAN, and our proposed AC-WGAN have been conducted to test the capabilities of our proposed model. Inference and interpolation tests are conducted as well. Our proposed model outperformed existing style transfer system in generation’s delicacy, similarity, and efficiency. Incorporated with WGAN, the model also demonstrated a strong ability in providing a truthful training indicator with its loss. Study also suggested that AC and pix2pix adaption hold huge significance in accelerating the training process. Research validates the viability of fulfilling Chinese character transformation with style transfer. With handwriting recognition network, fast, high-quality Chinese character style transfer forms the basis for instantaneous conversion between printed and written work. Optimization on algorithm may be another direction for further exploration.