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Classifiers

Classifiers. Given a feature representation for images, how do we learn a model for distinguishing features from different classes?. Decision boundary. Zebra. Non-zebra. Classifiers.

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Classifiers

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  1. Classifiers • Given a feature representation for images, how do we learn a model for distinguishing features from different classes? Decisionboundary Zebra Non-zebra

  2. Classifiers • Given a feature representation for images, how do we learn a model for distinguishing features from different classes? • Today: • Nearest neighbor classifiers • Linear classifiers: support vector machines • Later: • Boosting • Decision trees and forests • Deep neural networks (hopefully)

  3. Review: Nearest Neighbor Classifier

  4. Review: Nearest Neighbor Classifier • Assign label of nearest training data point to each test data point from Duda et al.

  5. Review: K-Nearest Neighbors • For a new point, find the k closest points from training data • Labels of the k points “vote” to classify k = 5

  6. Distance functions for bags of features • Euclidean distance: • L1 distance: • χ2 distance: • Histogram intersection (similarity): • Hellinger kernel (similarity):

  7. Review: Linear classifiers

  8. Review: Linear classifiers • Find linear function (hyperplane) to separate positive and negative examples Which hyperplaneis best?

  9. Support vector machines • Find hyperplane that maximizes the margin between the positive and negative examples C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

  10. Support vector machines • Find hyperplane that maximizes the margin between the positive and negative examples For support vectors, Distance between point and hyperplane: Therefore, the margin is 2 / ||w|| Support vectors Margin C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

  11. Finding the maximum margin hyperplane • Maximize margin 2 / ||w|| • Correctly classify all training data: • Quadratic optimization problem: C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

  12. Finding the maximum margin hyperplane • Solution: Learned weight (nonzero only for support vectors) C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

  13. Finding the maximum margin hyperplane • Solution:b = yi – w·xi for any support vector • Classification function (decision boundary): • Notice that it relies on an inner product between the testpoint x and the support vectors xi • Solving the optimization problem also involvescomputing the inner products xi· xjbetween all pairs oftraining points C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

  14. What if the data is not linearly separable? • Separable: • Non-separable: • C: tradeoff constant, ξi: slack variable (positive) • Whenever margin is ≥ 1, ξi = 0 • Whenever margin is < 1,

  15. What if the data is not linearly separable? Maximize margin Minimize classification mistakes

  16. What if the data is not linearly separable? • Demo: http://cs.stanford.edu/people/karpathy/svmjs/demo +1 0 Margin -1

  17. x 0 x 0 x2 Nonlinear SVMs • Datasets that are linearly separable work out great: • But what if the dataset is just too hard? • We can map it to a higher-dimensional space: 0 x Slide credit: Andrew Moore

  18. Nonlinear SVMs • General idea: the original input space can always be mapped to some higher-dimensional feature space where the training set is separable: Φ: x→φ(x) Slide credit: Andrew Moore

  19. Nonlinear SVMs • The kernel trick: instead of explicitly computing the lifting transformation φ(x), define a kernel function K such thatK(x,y) = φ(x)· φ(y) • (to be valid, the kernel function must satisfy Mercer’s condition) • This gives a nonlinear decision boundary in the original feature space: C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998

  20. x2 Nonlinear kernel: Example • Consider the mapping

  21. Polynomial kernel:

  22. Gaussian kernel • Also known as the radial basis function (RBF) kernel: • The corresponding mapping φ(x)is infinite-dimensional!

  23. Gaussian kernel SV’s

  24. Gaussian kernel • Also known as the radial basis function (RBF) kernel: • The corresponding mapping φ(x)is infinite-dimensional! • What is the role of parameter σ? • What if σ is close to zero? • What if σ is very large?

  25. Kernels for bags of features • Histogram intersection kernel: • Hellinger kernel: • Generalized Gaussian kernel: • D can be L1, Euclidean, χ2distance, etc. J. Zhang, M. Marszalek, S. Lazebnik, and C. Schmid, Local Features and Kernels for Classifcation of Texture and Object Categories: A Comprehensive Study, IJCV 2007

  26. Summary: SVMs for image classification • Pick an image representation (in our case, bag of features) • Pick a kernel function for that representation • Compute the matrix of kernel values between every pair of training examples • Feed the kernel matrix into your favorite SVM solver to obtain support vectors and weights • At test time: compute kernel values for your test example and each support vector, and combine them with the learned weights to get the value of the decision function

  27. What about multi-class SVMs? • Unfortunately, there is no “definitive” multi-class SVM formulation • In practice, we have to obtain a multi-class SVM by combining multiple two-class SVMs • One vs. others • Traning: learn an SVM for each class vs. the others • Testing: apply each SVM to test example and assign to it the class of the SVM that returns the highest decision value • One vs. one • Training: learn an SVM for each pair of classes • Testing: each learned SVM “votes” for a class to assign to the test example

  28. SVMs: Pros and cons • Pros • Many publicly available SVM packages:http://www.kernel-machines.org/software • Kernel-based framework is very powerful, flexible • SVMs work very well in practice, even with very small training sample sizes • Cons • No “direct” multi-class SVM, must combine two-class SVMs • Computation, memory • During training time, must compute matrix of kernel values for every pair of examples • Learning can take a very long time for large-scale problems

  29. SVMs for large-scale datasets • Efficient linear solvers • LIBLINEAR, PEGASOS • Explicit approximate embeddings: define an explicit mapping φ(x)such that φ(x)· φ(y) approximatesK(x,y)and train a linear SVM on top of that embedding • Random Fourier features for the Gaussian kernel (Rahimi and Recht, 2007) • Embeddings for additive kernels, e.g., histogram intersection (Maji et al., 2013, Vedaldi and Zisserman, 2012)

  30. Summary: Classifiers • Nearest-neighbor and k-nearest-neighbor classifiers • Support vector machines • Linear classifiers • Margin maximization • Non-separable case • The kernel trick • Multi-class SVMs • Large-scale SVMs • Of course, there are many other classifiers out there • Neural networks, boosting, decision trees/forests, …

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