1. Pawan Lingras and Cory Butz
Interval Set Representations of 1-v-r Support Vector Machine Multi-classifiers
Pawan Lingras and Cory Butz

2. Figure 1. [5] Linear separable sample2
Figure 1. [5] Linear separable sample w b

3. 1 1 2 1 2 2 2 1 2 2 2 2 1 1 1 1 1

5. 1 2
Figure 3. [5] Maximizing the margin between two classes g
Minimize
such that

6. An equivalence class approximation Lower Actual set
Upper approximation
Actual set
Lower
approximation
Figure 4. Rough Sets

7. Rough sets

8. Rough Sets for SVM Binary Classification
Ideal scenario:
the transformed feature space is linearly separable
SVM has found the optimal hyperplane by maximizing the margin between the two classes
There are no examples in the margin

9. The optimal hyperplane gives us the best possible dividing line
makes no assumptions about the classification of objects in the margin
the margin will be boundary region
create rough sets as follows.

10. Rough Sets for SVM Binary Classification

11. 1 2
Figure 3. [5] Maximizing the margin between two classes g b1 b2

13. Problems of High dimensionality
Cristianini list disadvantages of refining feature space to achieve linear separability.
Often this will lead to high dimensions, which will significantly increase the computational requirements
it is easy to overfit in high dimensional spaces
regularities could be found in the training set that are accidental, which would not be found again in a test set.
The soft margin classifiers [5] modify the optimization problem to allow for an error rate.

14. 1-v-r SVM Multi-classification
Construct a binary SVM for each class
Objects are labeled as belonging to that class or not
N SVMs for N classes
One disadvantage
Sample size for each SVM is large

15. Rough Set Advantages for Soft Margins

16. Rough set based 1-v-r

20. Main Advantage of Rough set based 1-v-r
Storage and operational phase time requirements are the same as 1-v-r, namely, O(N)
Sample sizes are lower than 1-v-r

21. Conclusions
Rough set view useful for practical applications with soft margins
The 1-v-r extension has linear storage and operational time requirements
Reduced sample size advantage is realized in the 1-v-r approach