Relationship Matrix Nonnegative Decomposition for Clustering

Algorithm 1

Relationship matrix nonnegative decomposition (RMND).

Input: Positive matrix 𝑋∈ℛ^𝑛×𝑛, and a positive integer 𝑞.

Output: Nonnegative factor matrices 𝐴∈ℛ^𝑛×𝑞, 𝑀∈ℛ^𝑛×𝑞 and 𝐻∈ℛ^𝑞×𝑛.

Learning Procedure

 (S1) Initialize 𝐴, 𝑀 and 𝐻 to random positive matrices, and normalize each column of 𝐴 to one.

 (S2) Repeat the iterations until convergence:

  (1) 𝐻←𝐻⊗(𝑀^𝑇𝐴𝐴^𝑇𝑋)⊘(𝑀^𝑇𝐴𝐴^𝑇𝐴𝐴^𝑇𝑀𝐻).

  (2) 𝑀←𝑀⊗(𝐴𝐴^𝑇𝑋𝐻^𝑇)⊘(𝐴𝐴^𝑇𝐴𝐴^𝑇𝑀𝐻𝐻^𝑇).

  (3) 𝑀←𝑀𝑈⁻¹ and 𝐻←𝑈𝐻 where 𝑈 is a diagonal matrix with 𝑢_𝑗𝑗=∑_𝑖𝑚_𝑖𝑗.

  (4) Repeatedly select small positive constant 𝜇_𝐴 until the objective function is decreased.

   (i) 𝐴∶=𝐴−𝜇_𝐴𝜕𝐷𝜕𝐴.

   (ii) Project each column of 𝐴 to be nonnegative vector with unit 𝐿₂ norm.

  (5) 𝐴∶=𝐴.

  Above, ⊗ and ⊘ denote elementwise multiplication and division, respectively.