Abstract
We propose a graph model for mutual information based clustering problem. This problem was originally formulated as a constrained optimization problem with respect to the conditional probability distribution of clusters. Based on the stationary distribution induced from the problem setting, we propose a function which measures the relevance among data objects under the problem setting. This function is utilized to capture the relation among data objects, and the entire objects are represented as an edge-weighted graph where pairs of objects are connected with edges with their relevance. We show that, in hard assignment, the clustering problem can be approximated as a combinatorial problem over the proposed graph model when data is uniformly distributed. By representing the data objects as a graph based on our graph model, various graph based algorithms can be utilized to solve the clustering problem over the graph. The proposed approach is evaluated on the text clustering problem over 20 Newsgroup and TREC datasets. The results are encouraging and indicate the effectiveness of our approach.
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Notes
Probabilistic assignment of data object into several clusters is called soft assignment.
Minimizing − I(T;Y) is equivalent to maximizing I(T;Y).
Note that D KL [p(y|x) || p(y|t)] is not symmetric.
Each vertex has at least one edge with positive weight. For disconnected graphs, each component can be dealt with separately.
\(\sum_{x_j}\) ranges over \({\boldsymbol{X}}\) and corresponds to ∑ j .
\( I(X;Y) - I(T;Y) = \sum_{x,y} p(x,y) \log \frac{p(y|x)}{p(y)} - \sum_{y,t} p(y,t) \log \frac{p(y|t)}{p(y)} = \sum_{x,y,t} p(x,y,t) \left(\log \frac{p(y|x)}{p(y|t)} +\right.\)\( \left.\log \frac{p(y|t)}{p(y)}\right) - \sum_{x,y,t} p(x,y,t) \log \frac{p(y|t)}{p(y)} = \sum_{x} \sum_{t} p(x)p(t|x) \sum_{y} p(y|x) \log \frac{p(y|x)}{p(y|t)} = \sum_{x} \sum_{t} p(x)p(t|x)\)D KL [p(y|x) || p(y|t)]
\(\bar S\) is the complement of S. We follow the convention to utilize the symbol S to denote the subset in a partition.
S and \(\bar S\) corresponds to clusters.
Any hard assignment deviates from (6).
Although it is possible to deal with asymmetric matrix, we focus on symmetric one in this paper.
l corresponds to the number of dimension of the embedded subspace.
A dataset for new3 contains 2,200 data items. One run of iIB took more than 3 h, and we could not evaluate 100 runs for each value of β.
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Acknowledgements
We express sincere gratitude to the reviewers for their careful reading of the manuscript and for providing valuable suggestions to improve the paper. This work is partially supported by the grant-in-aid for scientific research (No. 20500123) funded by MEXT, Japan.
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Yoshida, T. A graph model for mutual information based clustering. J Intell Inf Syst 37, 187–216 (2011). https://doi.org/10.1007/s10844-010-0132-5
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DOI: https://doi.org/10.1007/s10844-010-0132-5