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Bi-Clustering (I): The δ-Cluster Algorithm

  • Maximal δ-bi-cluster is a δ-bi-cluster I x J such that there does not exist another δ-bi-cluster I′ x J′ which contains I x J
  • Computing is costly: Use heuristic greedy search to obtain local optimal clusters
  • Two phase computation: deletion phase and additional phase
  • Deletion phase: Start from the whole matrix, iteratively remove rows and columns while the mean squared residue of the matrix is over δ
    • At each iteration, for each row/column, compute the mean squared residue:

\[d(i)=\frac{1}{J}\sum_{j\epsilon J}(e_{ij}-e_{iJ}-e_{Ij}+e_{IJ})^{2}\]

\[d(j)=\frac{1}{I}\sum_{i\epsilon I}(e_{ij}-e_{iJ}-e_{Ij}+e_{IJ})^{2}\]

    • Remove the row or column of the largest mean squared residue
  • Addition phase:
    • Expand iteratively the δ-bi-cluster I x J obtained in the deletion phase as long as the δ-bi-cluster requirement is maintained
    • Consider all the rows/columns not involved in the current bi-cluster I x J by calculating their mean squared residues
    • A row/column of the smallest mean squared residue is added into the current δ-bi-cluster
  • It finds only one δ-bi-cluster, thus needs to run multiple times: replacing the elements in the output bi-cluster by random numbers

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