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

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

1 1 2 1 2 2 2 1 2 2 2 2 1 1 1 1 1

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

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

Rough sets

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

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.

Rough Sets for SVM Binary Classification

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

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.

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

Rough Set Advantages for Soft Margins

Rough set based 1-v-r

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

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