• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.59-80
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• 2003

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.

• Journal of the Korean Physical Society
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• v.73 no.9
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• pp.1230-1239
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• 2018

We derive a factorization theorem for the jet mass distribution with a given $p^J_T$ for the inclusive production, where $p^J_T$ is a large jet transverse momentum. Considering the small jet radius limit ($R{\ll}1$), we factorize the scattering cross section into a partonic cross section, the fragmentation function to a jet, and the jet mass distribution function. The decoupled jet mass distributions for quark and gluon jets are well-normalized and scale invariant, and they can be extracted from the ratio of two scattering cross sections such as $d{\sigma}/(dp^J_TdM^2_J)$ and $d{\sigma}/dp^J_T $. When $M_J{\sim}p^J_TR$, the perturbative series expansion for the jet mass distributions works well. As the jet mass becomes small, large logarithms of $M_J/(p^J_TR)$ appear, and they can be systematically resummed through a more refined factorization theorem for the jet mass distribution.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.13 no.9
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• pp.4684-4705
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• 2019

Collaborative filtering recommender systems are vulnerable to shilling attacks in which malicious users may inject biased profiles to promote or demote a particular item being recommended. To tackle this problem, many robust collaborative recommendation methods have been presented. Unfortunately, the robustness of most methods is improved at the expense of prediction accuracy. In this paper, we construct a robust Bayesian probabilistic matrix factorization model for collaborative filtering recommender systems by incorporating the detection of user anomaly rating behaviors. We first detect the anomaly rating behaviors of users by the modified K-means algorithm and target item identification method to generate an indicator matrix of attack users. Then we incorporate the indicator matrix of attack users to construct a robust Bayesian probabilistic matrix factorization model and based on which a robust collaborative recommendation algorithm is devised. The experimental results on the MovieLens and Netflix datasets show that our model can significantly improve the robustness and recommendation accuracy compared with three baseline methods.

• East Asian mathematical journal
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• v.39 no.5
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• pp.523-528
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• 2023

Let S_n = [S(i, j)]_1≤i,j≤n and S^*_n = [S(i + j, j)]_1≤i,j≤n where S(i, j) is the Stirling number of the second kind. Choi and Jo [On the determinants of the square-type Stirling matrix and Bell matrix, Int. J. Math. Math. Sci. 2021] obtained the diagonal entries of matrix U in the LU-factorization of S^*_n for calculating the determinant of S^*_n, where L = S_n. In this paper, we compute the all entries of U in the LU-factorization of matrix S^*_n. This implies the identities related to Stirling numbers of both kinds.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.17 no.3
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• pp.721-739
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• 2023

Rating prediction is an important issue in recommender systems, and its accuracy affects the experience of the user and the revenue of the company. Traditional recommender systems use Factorization Machinesfor rating predictions and each feature is selected with the same weight. Thus, there are problems with inaccurate ratings and limited data representation. This study proposes a deep recommendation model based on self-attention Factorization (SAFMR) to solve these problems. This model uses Convolutional Neural Networks to extract features from user and item reviews. The obtained features are fed into self-attention mechanism Factorization Machines, where the self-attention network automatically learns the dependencies of the features and distinguishes the weights of the different features, thereby reducing the prediction error. The model was experimentally evaluated using six classes of dataset. We compared MSE, NDCG and time for several real datasets. The experiment demonstrated that the SAFMR model achieved excellent rating prediction results and recommendation correlations, thereby verifying the effectiveness of the model.

• Journal of KIISE:Software and Applications
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• v.36 no.5
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• pp.347-352
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• 2009

Nonnegative matrix factorization (NMF) is a popular method for multivariate analysis of nonnegative data, the goal of which is decompose a data matrix into a product of two factor matrices with all entries in factor matrices restricted to be nonnegative. NMF was shown to be useful in a task of clustering (especially document clustering). In this paper we present an algorithm for orthogonal nonnegative matrix factorization, where an orthogonality constraint is imposed on the nonnegative decomposition of a term-document matrix. We develop multiplicative updates directly from true gradient on Stiefel manifold, whereas existing algorithms consider additive orthogonality constraints. Experiments on several different document data sets show our orthogonal NMF algorithms perform better in a task of clustering, compared to the standard NMF and an existing orthogonal NMF.

• Journal of Digital Contents Society
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• v.18 no.4
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• pp.707-712
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• 2017

As the amount of data increases exponentially, the recommender system is attracting interest in various industries such as movies, books, and music, and is being studied. The recommendation system aims to propose an appropriate item to the user based on the user's past preference and click stream. Typical examples include Netflix's movie recommendation system and Amazon's book recommendation system. Previous studies can be categorized into three types: collaborative filtering, content-based recommendation, and hybrid recommendation. However, existing recommendation systems have disadvantages such as sparsity, cold start, and scalability problems. To improve these shortcomings and to develop a more accurate recommendation system, we have designed a recommendation system as a factorization machine using actual online product purchase data.

• Honam Mathematical Journal
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• v.23 no.1
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• pp.59-66
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• 2001

A class of topological algebras, which we call it a fundamental one, has already been introduced to generalize the locally bounded and locally convex algebras. To prove the basic theorems on fundamental algebras, the first successful step is the new version of the Cohen factorization theorem. Here we recall it and prove some new results on fundamental topological algebras.

• Journal of the Korean Mathematical Society
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• v.46 no.2
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• pp.281-294
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• 2009

Accuracy of the scaling function is very crucial in wavelet theory, or correspondingly, in the study of wavelet filter banks. We are mainly interested in vector-valued filter banks having matrix factorization and indicate how to choose block central symmetric matrices to construct multi-wavelets with suitable accuracy.

• The Journal of the Acoustical Society of Korea
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• v.34 no.3
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• pp.240-246
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• 2015

In this paper, speech enhancement using nonnegative matrix factorization with temporal continuity has been addressed. Speech and noise signals are modeled as Possion distributions, and basis vectors and gain vectors of NMF are modeled as Gamma distributions. Temporal continuity of the gain vector is known to be critical to the quality of enhanced speech signals. In this paper, temporal continiuty is implemented by adopting Gamma-Markov chain priors for noise gain vectors during the separation phase. Simulation results show that the Gamma-Markov chain models temporal continuity of noise signals and track changes in noise effectively.