TY - GEN
T1 - Mining discrete patterns via binary matrix factorization
AU - Shen, Bao Hong
AU - Ji, Shuiwang
AU - Ye, Jieping
PY - 2009
Y1 - 2009
N2 - Mining discrete patterns in binary data is important for sub- sampling, compression, and clustering. We consider rank- one binary matrix approximations that identify the dominant patterns of the data, while preserving its discrete property. A best approximation on such data has a minimum set of inconsistent entries, i.e., mismatches between the given binary data and the approximate matrix. Due to the hardness of the problem, previous accounts of such problems employ heuristics and the resulting approximation may be far away from the optimal one. In this paper, we show that the rank-one binary matrix approximation can be reformulated as a 0-1 integer linear program (ILP). However, the ILP formulation is computationally expensive even for small-size matrices. We propose a linear program (LP) relaxation, which is shown to achieve a guaranteed approximation error bound. We further extend the proposed formulations using the regularization technique, which is commonly employed to address overfitting. The LP formulation is restricted to medium-size matrices, due to the large number of variables involved for large matrices. Interestingly, we show that the proposed approximate formulation can be transformed into an instance of the minimum s-t cut problem, which can be solved efficiently by finding maximum flows. Our empirical study shows the efficiency of the proposed algorithm based on the maximum flow. Results also confirm the established theoretical bounds.
AB - Mining discrete patterns in binary data is important for sub- sampling, compression, and clustering. We consider rank- one binary matrix approximations that identify the dominant patterns of the data, while preserving its discrete property. A best approximation on such data has a minimum set of inconsistent entries, i.e., mismatches between the given binary data and the approximate matrix. Due to the hardness of the problem, previous accounts of such problems employ heuristics and the resulting approximation may be far away from the optimal one. In this paper, we show that the rank-one binary matrix approximation can be reformulated as a 0-1 integer linear program (ILP). However, the ILP formulation is computationally expensive even for small-size matrices. We propose a linear program (LP) relaxation, which is shown to achieve a guaranteed approximation error bound. We further extend the proposed formulations using the regularization technique, which is commonly employed to address overfitting. The LP formulation is restricted to medium-size matrices, due to the large number of variables involved for large matrices. Interestingly, we show that the proposed approximate formulation can be transformed into an instance of the minimum s-t cut problem, which can be solved efficiently by finding maximum flows. Our empirical study shows the efficiency of the proposed algorithm based on the maximum flow. Results also confirm the established theoretical bounds.
KW - Binary matrix factorization
KW - Integer linear program
KW - Maximum flow
KW - Minimum cut
KW - Rank-one
KW - Regularization
UR - http://www.scopus.com/inward/record.url?scp=70350623326&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=70350623326&partnerID=8YFLogxK
U2 - 10.1145/1557019.1557103
DO - 10.1145/1557019.1557103
M3 - Conference contribution
AN - SCOPUS:70350623326
SN - 9781605584959
T3 - Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining
SP - 757
EP - 765
BT - KDD '09
T2 - 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD '09
Y2 - 28 June 2009 through 1 July 2009
ER -