The Methods for Approximation of Principal Points for Binary Distributions on the Basis of Submodularity

An Algorithm for Principal Points Considering External Criterion for Multivariate Binary Distributions

“…Yamashita and Suzuki [2] proposed the following definition of principal points for binary distributions:…”

“…If the combinations of k candidates out of 2 p are considered, the number of their combinations increases exponentially with an increase in p. Therefore, this problem is regarded as a NP-hard problem. The algorithms proposed by Yamashita and Suzuki [2,3] were based on a k-means algorithm and a greedy algorithm, respectively.…”

“…For example, when we analyze multiple choice questionnaire survey data in which the number of selected items is predetermined, dummy variables data including categories of sex, blood type, and occupation, it is useful to find binary "principal" points where the number of occurrences l of response "1" is set. Yamashita and Suzuki [2] proposed a special definition for k-principal points of binary distributions when the number of occurrences l of response "1" is given, and these points were referred to as "l-k-principal points. "…”

“…For the algorithm of principal points for a multivariate binary distribution, Yamashita and Suzuki [2] proposed the greedy algorithm, and they showed that the approximations feature submodularity [6], based on the greedy algorithm, and guarantee (1 − 1/e) times the optimal solution. This algorithm can be adopted to solve the problem of principal points considering external criterion for multivariate binary distribution.…”

“…However, the objects of this analysis method are continuous values; thus, interpreting the analysis results can be quite difficult. For this reason, Yamashita and Suzuki [2] proposed using principal points for binary data. In particular, they proposed representing the data using a small number of realizations (i.e., a small number of binary points) of the multivariate binary distribution.…”