EVD Dualdating Based Online Subspace Learning

Algorithm 2

EVD dualdating.

Input: The rank-𝑘 EVD of old data 𝐴∈ℝ^𝑑×𝑛, 𝑈_𝑘, Λ_𝑘, the deleted data 𝐷∈ℝ^𝑑×𝑝, the added data 𝐵∈ℝ^𝑑×𝑚.

Output: The rank-𝑘 EVD of the remained data 𝐴^⋆∈ℝ^{𝑑×(𝑛+𝑚−𝑝)}, 𝑈^⋆_𝑘, Λ^⋆_𝑘.

 (1) Let equivalent adding data matrix 𝐵^⋆=𝐵𝐷.

 (2) Compute QR decomposition, (𝐼_𝑑−𝑈_𝑘𝑈^𝑇_𝑘)𝐵^⋆=QR.

 (3) Σ_𝑘=Λ_𝑘^1/2, compute the rank-𝑘 SVD, Σ_𝑘𝑈^𝑇_𝑘𝐵^⋆=𝑈0𝑅^𝑘Σ_𝑘𝑉^𝑇_𝑘.

 (4) Let 𝐼Φ=^𝑘𝐼_𝑚−𝐼_𝑝, calculate Ψ=Σ_𝑘𝑉^𝑇_𝑘Φ𝑉_𝑘Σ_𝑘.

 (5) Compute the EVD of Ψ, Ψ=𝑃Λ^⋆𝑃^𝑇.

 (6) The rank-𝑘 EVD of 𝐴^⋆ is

                 𝑈^⋆_𝑘=𝑈_𝑘𝑄𝑈_𝑘𝑃,Λ^⋆_𝑘=Λ^⋆.