Gene-Based Multiclass Cancer Diagnosis with Class-Selective Rejections

Algorithm 1

Multiclass SVM minimizing an asymmetric loss function.

1    𝜽∶=∅

2   𝐶∶=∅

3    for𝜎←𝜎₀ to 𝜎_𝑠 do

4         /^∗ Using the Training Set ∗ /

5         for 𝑗←1 to 𝑁 do

6    Train 𝜈-1-SVM on 𝑤_𝑗, namely solving the QP (16)

7    Derive the regularization path for 𝑤_𝑗, namely compute the 𝜶^𝜆𝑗s

8         end

9         /^∗ Using the Validation Set ∗ /

10     𝝀∶=𝝀₀

11     𝜷∶=𝜷₀

12     repeat

13         𝑑^𝜆𝑗1(𝑥)∶=𝜆_𝑗∑^𝑛𝑗_𝑝=1𝛼^𝜆𝑗_𝑝𝐾(𝑥_𝑝,𝑥)/𝑏^𝜆𝑗

14         𝑃_𝑗∶=|𝑤_𝑗|/∑^𝑁_𝑗=1|𝑤_𝑗| /^∗ || = cardinality∗ /

15         Assign 𝑥 to a decision 𝜓_𝑖 according to (19)

16         𝑃(𝐷_𝑖/𝑤_𝑗)∶=|{𝑥of𝑤_𝑗assignedto𝜓_𝑖}|/|{𝑥/𝑥∈𝑤_𝑗}|

17         𝑐(̂𝑍)∶=∑^𝐼_𝑖=1∑^𝑁_𝑗=1𝑐_𝑖𝑗𝑃_𝑗𝑃(𝐷_𝑖/𝑤_𝑗)

18       𝝀∶=𝝀_new /^∗ construct the new vector according to the

     direction of greatest decrease ∗ /

19       𝜷∶=𝜷_new

20     until 𝑐(̂𝑍)is minimum

21     𝜽∶=𝜽∪{𝜎,𝝀,𝜷}

22   𝐶∶=𝐶∪{𝑐(̂𝑍)}

23     end

24     index := min{𝐶}

25    𝜽_optimal∶=𝜽_index