Maximum Neighborhood Margin Discriminant Projection for Classification

Algorithm 1

Maximum neighborhood margin discriminant projection.

Require:

 𝑥∈ℝ^𝑚: a testing point. 𝑋={𝑥_𝑖∈ℝ^𝑚}^𝑁_𝑖=1: a training set.

Ensure:

 Predict the class label of 𝑥.

 Step  1. Construct the adjacent graph 𝒢 for any point

 in the training set 𝑋 using k-neighborhood.

 Step  2. Compute the affinity weight matrix 𝑊⁺

 for intraclass neighborhood and 𝑊⁻ for interclass

 neighborhood of any point, respectively.

 If  𝑥_𝑗∈𝒩⁺_𝑘(𝑥_𝑖) or 𝑥_𝑖∈𝒩⁺_𝑘𝑥_𝑗,  then

  𝑊⁺_𝑖𝑗−𝑥=exp^𝑖−𝑥_𝑗²𝛿−𝑥1+exp^𝑖−𝑥_𝑗²𝛿

 else

  0

 end if

 if  𝑥_𝑗∈𝒩⁻_𝑘(𝑥_𝑖) or 𝑥_𝑖∈𝒩⁻_𝑘𝑥_𝑗,  then

  𝑊⁻_𝑖𝑗−𝑥=exp^𝑖−𝑥_𝑗²𝛿−𝑥1−exp^𝑖−𝑥_𝑗²𝛿

 else

  0

 end if

 Step  3. Compute the intraclass neighborhood scatter

 matrix 𝑆⁺=𝑋𝐿⁺𝑋^𝑇 and the interclass neighborhood

 scatter matrix 𝑆⁻=𝑋𝐿⁻𝑋^𝑇.

 Step  4. Obtain the optimal projection matrix Φ

 by maximizing the generalized eigenvalue problem

 (𝑆⁻−𝑆⁺)Φ=𝜆Φ.

 Step  5. Dimensionality reduction: transform all the

 points from the high-dimensional feature space 𝑋 to

 a subspace 𝑌 with the optimized projection matrix Φ,

 that is, 𝑌=Φ^𝑇𝑋.

 Step  6. Classify 𝑥 using a certain classifier. The projection

 of 𝑥 is first obtained by 𝑦=Φ^𝑇𝑥 and then

 classify 𝑦 in the projected subspace 𝑌.