Fast Second-Order Orthogonal Tensor Subspace Analysis for Face Recognition

Algorithm 1

TSA.

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_𝐷^(𝑈)=𝑃^𝑇_𝑈𝐷⊗𝐼_𝑙1𝑃_𝑈;

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

   (2.3) Compute the 𝑙₂ eigenvectors {𝑣_𝑖}^𝑙2_𝑖=1 of the pencil 𝑀_𝐷^(𝑈),𝑀_𝐿^(𝑈)

     corresponding to the largest 𝑙₂ eigenvalues.

   (2.4) Set 𝑉=[𝑣₁,𝑣₂,…,𝑣_𝑙2];

   (2.5) Form the matrix 𝑀_𝐷^(𝑉)=𝑃^𝑇_𝑉𝐷⊗𝐼_𝑙2𝑃_𝑉;

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

   (2.7) Compute the 𝑙₁ eigenvectors {𝑢_𝑖}^𝑙1_𝑖=1 of the pencil 𝑀_𝐷^(𝑉),𝑀_𝐿^(𝑉)

     corresponding to the largest 𝑙₁ eigenvalues.

   (2.8) Set 𝑈=𝑢₁,𝑢₂,…,𝑢_𝑙1;

   End Do