Tensorial Kernel Principal Component Analysis for Action Recognition

Algorithm 1

TKPCA: Kernel Principal Component Analysis of tensor objects.

(1) Tensorial Kernel Function  𝑘(𝒳_𝑖,𝒳_𝑗):

(2) for 𝑛=1 to 𝑁  do

(3)   Compute orthonormal bases 𝐔_𝒳^(𝑛)_𝑖 𝐔_𝒳^(𝑛)_𝑗 from SVD: 𝒳_𝑖(𝑛)=𝐔_𝒳^(𝑛)_𝑖𝐃_𝒳^(𝑛)_𝑖𝐕^(𝑛)𝑇_𝒳𝑖,

    𝒳_𝑗(𝑛)=𝐔_𝒳^(𝑛)_𝑗𝐃_𝒳^(𝑛)_𝑗𝐕^(𝑛)𝑇_𝒳𝑗.

(4) Compute 𝑘(𝒳_𝑖,𝒳_𝑗) by (22).

(5) TKPCA:

(6) Step  1. (Initialization):

(7) Center the input samples by subtracting empirical mean: (1/𝑀)∑^𝑀_𝑚=1𝒳_𝑚.

(8) Step  2. (Training):

(9) for  𝑖=1 to 𝑀  do

(10)   for  𝑗=1 to 𝑀  do

(11)     if  𝑖≠𝑗  then

(12)      Compute the Kernel Matrix (10): 𝓚(𝑖,𝑗)=𝑘(𝒳_𝑖,𝒳_𝑗).

(13) Compute the 𝑃 dominant eigenvectors 𝜶_𝑝 by eign-decomposition 𝓚 (9).

(14) Step  3. (Projection):

(15) For test sample 𝒳^′, compute the 𝑃 projections onto each one of the dominant eigenvectors (11),

   𝑦_𝑝≡⟨𝐰,𝝓(𝒳^′)⟩=∑^𝑀_𝑚=1𝜶_𝑝(𝑚)𝑘(𝒳_𝑚,𝒳^′), 𝑝=1,…,𝑃.