1. CS558 Project Local SVM Classification based on triangulation (on the plane) Glenn Fung

2. Outline of Talk  Classification problem on the plane  All of the recommended stages were applied:  Sampling  Ordering:  Clustering  Triangulation  Interpolation (Classification)  SVM: Support vector Machines  Optimization: Number of training points increased  Evaluation:  Checkerboard dataset  Spiral dataset

3. Classification Problem in  Given m points in 2 dimensional space  Represented by an m-by-2 matrix A  Membership of each in class +1 or –1

4. SAMPLING: 1000 randomly sampled points

5. ORDERING: Clustering  A Fuzzy-logic based clustering algorithm was used.  32 cluster centers were obtained

6. ORDERING: Delaunay Triangulation  Algorithms to triangulate and to get the Delaunay triangulation from HWKs 3 and 4 were used.  Given a point,the random point approach is used to localize the triangle that contains it.

7. Interpolation: SVM  SVM : Support Vector Machine Classifiers  A different nonlinear Classifier is used for each triangle  The triangle structure is efficiently used for both training and testing phases and for defining a “simple” and fast nonlinear classifier.

8. What is a Support Vector Machine?  An optimally defined surface  Typically nonlinear in the input space  Linear in a higher dimensional space  Implicitly defined by a kernel function

9. What are Support Vector Machines Used For?  Classification  Regression & Data Fitting  Supervised & Unsupervised Learning (Will concentrate on classification)

10. Support Vector Machines Maximizing the Margin between Bounding Planes A+ A-

11. The Nonlinear Classifier  The nonlinear classifier:  Where K is a nonlinear kernel, e.g.:  Gaussian (Radial Basis) Kernel :  The -entry of represents the “similarity” of data pointsand

12. Reduced Support Vector Machine Algorithm Nonlinear Separating Surface: (i) Choose a random subset matrix of entire data matrix (ii) Solve the following problem by the Newton method with corresponding : min (iii) The separating surface is defined by the optimal solution in step (ii):

13. How to Choose in RSVM?  is a representative sample of the entire dataset  Need not be a subset of  A good selection of may generate a classifier using very small  Possible ways to choose :  Choose random rows from the entire dataset  Choose such that the distance between its rows exceeds a certain tolerance  Use k cluster centers of as and

15. Obtained Bizarre “Checkerboard”

16. Optimization: More sampled points Training parameters adjusted

17. Result: Improved Checkerboard

18. Nonlinear PSVM: Spiral Dataset 94 Red Dots & 94 White Dots

19. Next:Bascom Hill

20. Some Questions  Would it work for B&W pictures (regression instead of classification?  Aplications?