A Review of Subspace Segmentation: Problem, Nonlinear Approximations, and Applications to Motion Segmentation

Algorithm 2

Optimal solution 𝐒^𝑜.

1: Pick any partition 𝑃∈𝒫 (W)

2: For each subset 𝐖_𝑖 in the partition 𝑃 find the subspace 𝑆^𝑜_𝑖(𝑃)∈𝒞

 that minimizes the expression 𝑒(𝐖_𝑖,𝑆) = ∑_{𝑤∈𝐖𝑖}𝑑^𝑝(𝑤,𝑆)

3: while  ∑^𝑀_𝑖=1𝑒(𝐖_𝑖,𝑆^𝑜_𝑖(𝑃))>𝑒(𝐖,𝐒^𝑜(𝑃)) do

4:  for all i from 1 to 𝑀 do

5:   Update 𝐖_𝑖={𝑤∈𝐖∶𝑑𝑤,𝑆^𝑜_𝑖(𝑃)≤𝑑𝑤,𝑆^𝑜_𝑘(𝑃),𝑘=1,…,𝑀}

6:   Update 𝑆^𝑜_𝑖 (P) =argmin_𝑆∈𝒞𝑒(𝐖_𝑖,𝑆)

7:  end for

8:  Update 𝑃={𝐖₁,…,𝐖_𝑀}

9: end while

10: 𝐒^𝑜 = {𝑆^𝑜₁(𝑃), …,𝑆^𝑜_𝑀(𝑃)}