Low-rank Approximation Of Integer Matrix And Its Applications

With the rapid development of the information technology in modern society,a variety of financial data that people used are growing at an explosive speed.These large amount of data are stored in the databases,but it is hard to use them totally.We need to extract useful information from databases through means of analysis,which is a hot topic in data mining,machine learning,pattern recognition and so on.The purpose of data mining technology is to reveal implicit,unknown and potentially valuable information from the massive data in the database.Unlike data analysis,data mining is not only for the analysis of data and research needs,but also for the acquisition of truly valuable information to help the needs of all aspects of society.There are many data mining functions,such as:? Automatic estimation and prediction: data mining can classify research objects through valuation,and can also automatically find predictive information in the database,directly translate the data itself into the information of people's needs,and solve the difficulties of traditional manual analysis.? Association rules analysis: data mining technology can find out some hidden relationship among information from data.For example,market basket transaction case analyzes the customer's shopping behavior by discovering the relationship between different products purchased by the customer.This associated discovery can help sellers understand customers' consumption habits,help them develop better marketing strategies,and also bring convenience to people.? Clustering analysis: The requirements of clustering technology in data mining are that it can handle high-dimensional data;it can find data packets with good characteristics under various constraints;it also has interpretability and availability.? Concept feature description: summarizing the data of the object studied,describing the features of the object,extracting more potential information,such as model extraction and image recognition processing.Matrix decomposition is one of the most important research methods in data mining.In particularly,low rank approximation is an important means of extracting its characteristic information.Up to now,many researchers have made great efforts in real or nonnegative matrix approximation and obtained a lot of useful results.However,there is no much research on the integer matrix.Because integers are discrete in nature,to our knowledge,no previously proposed technique developed for real numbers can be successfully applied.In this study,we first conduct a thorough review of current algorithms that can solve integer least squares problems,and then develop an alternative least square method based on the integer least squares estimation to obtain the integer approximation of the integer matrices.We discuss numerical applications for the approximation of randomly generated integer matrices as well as studies of association rule mining,cluster analysis,and pattern extraction.Our numerical results suggest that our proposed method can calculate a more accurate solution for discrete datasets than other existing methods.