Fast Second-Order Orthogonal Tensor Subspace Analysis for Face Recognition

Algorithm 7

ODTSA.

Input: A set of 𝑁 sample matrices {𝑋_𝑖}^𝑁_𝑖=1 with class label information, 𝑙₁, 𝑙₂.

Output: left and right transformation matrices 𝑈 and 𝑉.

  (1) Initialize 𝑈 with an identity matrix;

  (2) Until convergence Do:

   (2.1) Form the matrix 𝑀_𝐿^(𝑈)_𝐵=𝑄^𝑇_𝑈𝐿_𝐵⊗𝐼_𝑙1𝑄_𝑈;

   (2.2) Form the matrix 𝑀_𝐿^(𝑈)_𝑊=𝑃^𝑇_𝑈𝐿_𝑊⊗𝐼_𝑙1𝑃_𝑈;

   (2.3) Compute 𝑉 by solving the trace ratio optimization problem (27) with

     𝐴=𝑀_𝐿^(𝑈)_𝐵 and 𝐵=𝑀_𝐿^(𝑈)_𝑊;

   (2.4) Form the matrix 𝑀_𝐿^(𝑉)_𝐵=𝑄^𝑇_𝑉𝐿_𝐵⊗𝐼_𝑙2𝑄_𝑉;

   (2.5) Form the matrix 𝑀_𝐿^(𝑉)_𝑊=𝑃^𝑇_𝑉𝐿_𝑊⊗𝐼_𝑙2𝑃_𝑉;

   (2.6) Compute 𝑈 by solving the trace ratio optimization problem (27) with

     𝐴=𝑀_𝐿^(𝑉)_𝐵 and 𝐵=𝑀_𝐿^(𝑉)_𝑊.

   End Do