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Multiple Kernel Learning

Multiple Kernel Learning. Manik Varma Microsoft Research India. A Quick Re view of SVMs. Margin = 2 /  w t w.  > 1. Misclassified point.  < 1. b. Support Vector.  = 0. Support Vector. w. w t x + b = -1.  = 0. w t x + b = 0. w t x + b = +1.

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Multiple Kernel Learning

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  1. Multiple Kernel Learning Manik Varma Microsoft Research India

  2. A Quick Review of SVMs Margin = 2 /wtw  > 1 • Misclassified point  < 1 b • Support Vector  = 0 • Support Vector w wtx + b = -1  = 0 wtx + b = 0 wtx + b = +1

  3. Primal P = Minw,,b½wtw + Ct • s. t. Y(Xtw + b1) 1 –  • 0 • Dual D = Max 1t – ½tYKY • s. t. 1tY = 0 • 0    C The C SVM Primal and Dual

  4. Primal P = Minxf0(x) • s. t. fi(x) 0 1  i  N • hi(x)= 0 1  i  M • Lagrangian L(x,,) = f0(x) + i ifi(x) + i ihi(x) • Dual D = Max,Minx L(x,,) • s. t.  0 Duality

  5. The Lagrange dual is always concave (even if the primal is not convex) and might be an easier problem to optimize • Weak duality : P  D • Always holds • Strong duality : P = D • Does not always hold • Usually holds for convex problems • Holds for the SVM QP Duality

  6. If strong duality holds, then for x*, * and * to be optimal the following KKT conditions must necessarily hold • Primal feasibility : fi(x*) 0 & hi(x*)= 0 for 1  i • Dual feasibility : *  0 • Stationarity : xL(x*, *,*) = 0 • Complimentary slackness : i*fi(x*)= 0 • If x+, + and + satisfy the KKT conditions for a convex problem then they are optimal Karush-Kuhn-Tucker (KKT) Conditions

  7. Linear : K(xi,xj) = xit-1xj • Polynomial : K(xi,xj) = (xit-1xj + c)d • Gaussian (RBF) : K(xi,xj) = exp( –kk(xik – xjk)2) • Chi-Squared : K(xi,xj) = exp( –2(xi, xj) ) • Sigmoid : K(xi,xj) = tanh(xitxj – c) •  should be positive definite, c  0,   0 and d should be a natural number Some Popular Kernels

  8. Improve accuracy and generalization • Learn an RBF Kernel : K(xi,xj) = exp( – k (xik – xjk)2) Advantages of Learning the Kernel

  9. Kernel Parameter Setting - Underfitting

  10. Kernel Parameter Setting

  11. Kernel Parameter Setting – Overfitting

  12. Improve accuracy and generalization • Learn an RBF Kernel : K(xi,xj) = exp( – k (xik – xjk)2) • Test error as a function of  Advantages of Learning the Kernel

  13. Perform non-linear feature selection • Learn an RBF Kernel : K(xi,xj) = exp(–kk(xik – xjk)2) • Perform non-linear dimensionality reduction • Learn K(Pxi, Pxj) where P is a low dimensional projection matrix parameterized by  • These are optimized for the task at hand such as classification, regression, ranking, etc. Advantages of Learning the Kernel

  14. Multiple Kernel Learning • Learn a linear combination of given base kernels • K(xi,xj) = kdkKk(xi,xj) • Can be used to combine heterogeneous sources of data • Can be used for descriptor (feature) selection Advantages of Learning the Kernel

  15. MKL learns a linear combination of base kernels • K(xi,xj) = kdkKk(xi,xj) MKL – Geometric Interpretation d11 d22  = d33

  16. Suppose we’re given a simplistic 1D shape feature for a binary classification problem • Define a linear shape kernel : Ks(si,sj) = sisj • The classification accuracy is 100% but the margin is very small MKL – Toy Example s

  17. Suppose we’re now given addition 1D colour feature • Define a linear colour kernel : Kc(ci,cj) = cicj • The classification accuracy is also 100% but the margin remains very small MKL – Toy Example c

  18. MKL learns a combined shape-colour feature space • K(xi,xj) = dKs(xi,xj) + (1 – d) Kc(xi,xj) MKL – Toy Example c c s s d = 0 d = 1

  19. MKL – Toy Example

  20. MKL learns a combined shape-colour feature space • K(xi,xj) = dKs(xi,xj) + (1 – d) Kc(xi,xj) MKL – Another Toy Example c c s s d = 0 d = 1

  21. MKL – Another Toy Example

  22. Chair Object Categorization Schooner ? Ketch Taj Panda

  23. Database collected by Fei-Fei et al. [PAMI 2006] The Caltech 101 Database

  24. The Caltech 101 Database – Chairs

  25. The Caltech 101 Database – Bikes

  26. Features • Geometric Blur [Berg and Malik, CVPR 01] • PHOW Gray & Colour [Lazebnik et al., CVPR 06] • Self Similarity [Shechtman and Irani, CVPR 07] • Kernels • RBF for Geometric Blur • K(xi,xj) = exp( – 2(xi,xj)) for the rest Caltech 101 – Features and Kernels

  27. Experimental Setup • 102 categories including Background_Google and Faces_easy • 15 training and 15 test images per category • 30 training and up to 15 test images per category • Results summarized over 3 random train/test splits Caltech 101 – Experimental Setup

  28. Caltech 101 – MKL Results

  29. Caltech 101 – Comparisons

  30. Caltech 101 – Over Fitting?

  31. Caltech 101 – Over Fitting?

  32. Caltech 101 – Over Fitting?

  33. Caltech 101 – Over Fitting?

  34. Caltech 101 – Over Fitting?

  35. Experimental Setup • 33 topics chosen each with more than 60 images • Ntrain = [10, 15, 20, 25, 30] • The remaining images are used for testing • Features • PHOG 180 & 360 • Self Similarity • PHOW Gray & Colour • Gabor filters • Kernels • Pyramid Match Kernel & Spatial Pyramid Kernel Wikipedia MM Subset

  36. LMKL [Gonen and Alpaydin, ICML 08] • GS-MKL [Yang et al., ICCV 09] Wikipedia MM Subset

  37. FERET faces [Moghaddam and Yang, PAMI 2002] Feature Selection for Gender Identification Males Females

  38. Experimental setup • 1053 training and 702 testing images • We define an RBF kernel per pixel (252 kernels) • Results summarized over 3 random train/test splits Feature Selection for Gender Identification

  39. Feature Selection Results Uniform MKL = 92.6  0.9 Uniform GMKL = 94.3  0.1

  40. Localize a specified object of interest if it exists in a given image Object Detection

  41. The PASCAL VOC Challenge Database

  42. PASCAL VOC Database – Cars

  43. PASCAL VOC Database – Dogs

  44. PASCAL VOC 2009 Database Statistics

  45. Detect by classifying every image window at every position, orientation and scale • The number of windows in an image runs into the hundred millions • Even if we classify a window in a second it will take us many days to detect a single object in an image Bird Detection By Classification No Bird

  46. Fast Linear SVM Jumping Window Quasi-linear SVM Feature vector Non-linear SVM PHOW Gray Fast Detection Via a Cascade PHOW Colour PHOG PHOG Sym Visual Words Self Similarity

  47. First stage • Linear SVM • Jumping windows/Branch and Bound • Time = O(#Windows) • Second stage • Quasi-linear SVM • 2 kernel • Time = O(#Windows * #Dims) • Third stage • Non-linear SVM • Exponential 2 kernel • Time = O(#Windows * #Dims * #SVs) • Th MKL Detection Overview

  48. Predictions are evaluated using precision-recall curves based on bounding box overlap • Area Overlap = BgtBp / BgtBp • Valid prediction if Area Overlap > ½ PASCAL VOC Evaluation Ground truth Bgt BgtBp Predicted Bp

  49. Some Examples of MKL Detections

  50. Some Examples of MKL Detections

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