Direct Neighborhood Discriminant Analysis for Face Recognition

Algorithm 1

DNDA algorithm.

Input: Data matrix 𝑋∈𝑅^𝑑×𝑁, class label L

Output: Transformed matrix 𝑃^∗

1. Construct the between-class and the within-class affinity weight matrix 𝑊^𝑏, 𝑊^𝑤.

2. Construct the interclass and the intraclass index matrix H_s, H_c according to the nonzero elements

of 𝑊^𝑏, 𝑊^𝑤.

For the kth nonzero element of 𝑊^𝑏(𝑊^𝑤),𝑊^𝑏_𝑖𝑗(𝑊^𝑤_𝑖𝑗), the corresponding kth column in 𝐻_𝑠(𝐻_𝑐) is

constructed as

[0⋯0^{𝑖−10𝑥0𝑓𝑒37},1^𝑖,0⋯0,−^𝑗1,0⋯0^{𝑁−𝑗0𝑥0𝑓𝑒37}]^𝑇.

3. Apply eigenvalue decomposition to S_s and keep the largest t nonzero eigenvalues

𝜆=[𝜆₁,𝜆₂,…,𝜆_𝑡] and corresponding eigenvectors 𝑈=[𝑢₁,𝑢₂,…,𝑢_𝑡] after sorted in decreasing

order, where 𝑡=rank(𝑆_𝑠).

4. Compute P_s as 𝑃_𝑠=𝑈𝐷_𝑠^−1/2, where 𝐷_𝑠=diag(𝜆₁,𝜆₂,…,𝜆_𝑡) is diagonal matrix with 𝜆_𝑖 on the

main diagonal.

5. Perform eigenvalue decomposition on ^0𝑥0𝑆^02𝑑𝑐_𝑐=𝑃^𝑇_𝑠𝑆_𝑐𝑃_𝑠. Let 𝐷_𝑐=diag(𝜇₁,𝜇₂,…,𝜇_𝑛) be the

eigenvalue matrix of ^0𝑥0𝑆^02𝑑𝑐_𝑐 in ascending order and 𝑉=[𝑣₁,𝑣₂,…,𝑣_𝑛] be the corresponding

eigenvector matrix. Calculate P_c as 𝑃_𝑐=𝑉𝐷_𝑐^−1/2.

6. 𝑃^∗←𝑃_𝑠𝑃_𝑐.