1. Bioinformatics Challenge  Learning in very high dimensions with very few samples  Acute leukemia dataset: 7129 # of gene vs. 72 samples  Colon cancer dataset: 2000 # of gene vs. 62 samples  Feature selection will be needed

2. Feature Selection Approach Filter model  Weight score approach Wrapper model  1-norm SVM  IRSVM

3. Feature Selection – Filter Model Using Weight Score Approach Feature 1Feature 2Feature 3

4. Filter Model – Weight Score Approach Weight score: whereandare the mean and standard deviation of feature for training examples of positive or negative class.

5. Filter Model – Weight Score Approach is defined as the ratio between the difference of the means of expression levels and the sum of standard deviation in two classes. Selecting genes with largest as our top features. The weight score is calculated with the information about a single feature. The highly linear correlated features might be selected by this approach.

6. (Different Measure of Margin) 1-Norm SVM: 1-Norm SVM Equivalent to: Good for feature selection!

7. Clustering Process: Feature Selection & Initial Cluster Centers  6 out of 31 features selected by a linear SVM ( )  mean of area, standard error of area, worst area, worst texture, worst perimeter and tumor size

8. Reduced Support Vector Machine (ii) Solve the following problem by the Newton’s method min (iii) The nonlinear classifier is defined by the optimal solution in step (ii): Using gives lousy results! (i) Choose a random subset matrixof entire data matrix Nonlinear Classifier:

9. Reduced Set: Plays the Most Important Role in RSVM  It is natural to raise two questions:  Is there a way to choose the reduced set other than random selection so that RSVM will have a better performance?  Is there a mechanism to determine the size of reduced set automatically or dynamically?

10. Reduced Set Selection According to the Data Scatter in Input Space  Expected these points to be representative sample  Choose reduced set randomly but only keep the points in the reduced set that are more than a certain minimal distance apart

11. 1 2 3 5 4 6 7 8 9 11 10 12 A Better Way According to the Data Scatter in Feature Space  An example is given as following : Training data analogous to XOR problem

12. Mapping to Feature Space  Map the input data via nonlinear mapping ：  Equivalent to polynomial kernel with degree 2:

13. Data Points in the Feature Space 1 2 3 5 4 6 7 8 9 11 10 12 36 25 14 8 11 9 12 7 10

14. The Polynomial Kernel Matrix

15. 1 2 3 5 4 6 7 8 9 11 10 12 Experiment Result

16. Mathematical Observations Another Reason for IRSVM is a linear combination of a set of kernel functions  If the kernel functions are very similar, the hypothesis space spanned by this kernel functions will be very limited.  In SVMs, the nonlinear separating surface is:  In RSVMs, the nonlinear separating surface

17. Incremental Reduced SVMs The strength of weak ties  Start with a very small reduced set, then add a new data point only when the kernel vector is dissimilar to the current function set  This point contributes the most extra information for generating the separating surface  Repeat until several successive points cannot be added  The strength of weak ties (….)

18.  The distance from the kernel vector to the column space of is greater than a threshold  The criterion for adding a point into reduced set is  This distance can be determined by solving a least squares problem How to measure the dissimilar? Solving Least Squares Problems  It has a unique solution, and the distance is

19. IRSVM Algorithm pseudo-code (sequential version) 1 Randomly choose two data from the training data as the initial reduced set 2 Compute the reduced kernel matrix 3 For each data point not in the reduced set 4 Computes its kernel vector 5 Computes the distance from the kernel vector 6 to the column space of the current reduced kernel matrix 7 If its distance exceed a certain threshold 8 Add this point into the reduced set and form the new reduced kernel matrix 9 Until several successive failures happened in line 7 10 Solve the QP problem of nonlinear SVMs with the obtained reduced kernel 11 A new data point is classified by the separating surface

20. Wrapper Model – IRSVM Find a Linear Classifier: I. Randomly choose a very small feature subset from the input features as the initial feature reduced set. II. Select a feature vector not in the current feature reduced set and computing the distance between this vector and the space spanned by current feature reduced set. III. If the distance is larger than a given gap, then we add this feature vector to the feature reduced set. IV. Repeat step II and step III until there are no feature can be added to the current feature reduced set. V. Features in the resulting feature reduced set is our final result of feature selection.