Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

• Published in: Machine Learning and Knowledge Discovery in Databases Vol. 9285; pp. 36 - 52
• Main Author: Miettinen, Pauli
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2015; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Artificial intelligence; Binary Matrix; Data mining; Frequent Itemset; Matrix Factorization; Nonnegative Matrix Factorization; Pattern recognition; Unify Framework
• ISBN: 9783319235240; 3319235249
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-23525-7_3

Matrix factorizations are a popular tool to mine regularities from data. There are many ways to interpret the factorizations, but one particularly suited for data mining utilizes the fact that a matrix product can be interpreted as a sum of rank-1 matrices. Then the factorization of a matrix becomes the task of finding a small number of rank-1 matrices, sum of which is a good representation of the original matrix. Seen this way, it becomes obvious that many problems in data mining can be expressed as matrix factorizations with correct definitions of what a rank-1 matrix and a sum of rank-1 matrices mean. This paper develops a unified theory, based on generalized outer product operators, that encompasses many pattern set mining tasks. The focus is on the computational aspects of the theory and studying the computational complexity and approximability of many problems related to generalized matrix factorizations. The results immediately apply to a large number of data mining problems, and hopefully allow generalizing future results and algorithms, as well.