The Discrete Basis Problem

Matrix decomposition methods represent a data matrix as a product of two factor matrices: one containing basis vectors that represent meaningful concepts in the data, and another describing how the observed data can be expressed as combinations of the basis vectors. Decomposition methods have been s...

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Vydáno v:IEEE transactions on knowledge and data engineering Ročník 20; číslo 10; s. 1348 - 1362
Hlavní autoři: Miettinen, P., Mielikainen, T., Gionis, A., Das, G., Mannila, H.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York, NY IEEE 01.10.2008
IEEE Computer Society
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Algorithms
and association rules
Applied sciences
Artificial intelligence
Association rules
Boolean algebra
classification
Classification algorithms
Clustering
Clustering algorithms
Computer languages
Computer science; control theory; systems
Data processing. List processing. Character string processing
Decomposition
Exact sciences and technology
Greedy algorithms
Mathematical analysis
Matrices
Matrix decomposition
Matrix methods
Memory organisation. Data processing
Mining methods and algorithms
Operating systems
Partitioning
Partitioning algorithms
Program processors
Software
Speech and sound recognition and synthesis. Linguistics
Studies
Text mining
Vectors (mathematics)
Text mining
Clustering
classification
Mining methods and algorithms
and association rules
Cluster analysis
Matrix product
Statistical association
Discrete data
Matrix decomposition
Data mining
Boolean logic
Zero one matrix
Classification
NP hard problem
Greedy algorithm
Observation data
Matrix method
clustering
text mining
ISSN:1041-4347, 1558-2191
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Popis
Shrnutí:Matrix decomposition methods represent a data matrix as a product of two factor matrices: one containing basis vectors that represent meaningful concepts in the data, and another describing how the observed data can be expressed as combinations of the basis vectors. Decomposition methods have been studied extensively, but many methods return real-valued matrices. Interpreting real-valued factor matrices is hard if the original data is Boolean. In this paper, we describe a matrix decomposition formulation for Boolean data, the Discrete Basis Problem. The problem seeks for a Boolean decomposition of a binary matrix, thus allowing the user to easily interpret the basis vectors. We also describe a variation of the problem, the Discrete Basis Partitioning Problem. We show that both problems are NP-hard. For the Discrete Basis Problem, we give a simple greedy algorithm for solving it; for the Discrete Basis Partitioning Problem we show how it can be solved using existing methods. We present experimental results for the greedy algorithm and compare it against other, well known methods. Our algorithm gives intuitive basis vectors, but its reconstruction error is usually larger than with the real-valued methods. We discuss about the reasons for this behavior.
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ISSN:1041-4347
1558-2191
DOI:10.1109/TKDE.2008.53