Kernel Methods. Barnabás Póczos
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1 Kernel Methods Barnabás Póczos
2 Outline Quick Introduction Feature space Perceptron in the feature space Kernels Mercer s theorem Finite domain Arbitrary domain Kernel families Constructing new kernels from kernels Constructing feature maps from kernels Reproducing Kernel Hilbert Spaces (RKHS) The Representer Theorem
3 Ralf Herbrich: Learning Kernel Classifiers Chapter 3
4 Quick Overview
5 Hard 1-dimensional Dataset If the data set is not linearly separable, then adding new features (mapping the data to a larger feature space) the data might become linearly separable x=0 Positive plane Negative plane m general! points in an m-1 dimensional space is always linearly separable by a hyperspace! ) it is good to map the data to high dimensional spaces (For example 4 points in 3D) taken from Andrew W. Moore 5
6 Hard 1-dimensional Dataset Make up a new feature! Sort of computed from original feature(s) z k ( x k, x k ) Separable! MAGIC! x=0 Now drop this augmented data into our linear SVM. taken from Andrew W. Moore 6
7 Feature mapping m general! points in an m-1 dimensional space is always linearly separable by a hyperspace! ) it is good to map the data to high dimensional spaces Having m training data, is it always enough to map the data into a feature space with dimension m-1? Nope... We have to think about the test data as well! Even if we don t know how many test data we have... We might want to map our data to a huge (1) dimensional feature space Overfitting? Generalization error?... We don t care now... 7
8 Feature mapping, but how??? 1 8
9 Observation Several algorithms use the inner products only, but not the feature values!!! E.g. Perceptron, SVM, Gaussian Processes... 9
10 The Perceptron 10
11 Maximize k R k 1 k1 Subject to these constraints: R k1 R l1 SVM k l Q kl where 0 k C k Q kl y k y l (x k x l ) R k y k 0 k1 11
12 Inner products So we need the inner product between and Looks ugly, and needs lots of computation... Can t we just say that let 1
13 Finite example r r = n n r 13
14 Finite example Lemma: Proof: 14
15 3 Finite example 6 5 Choose 7 D points Choose a kernel k G =
16 [U,D]=svd(G), UDU T =G, UU T =I U = D =
17 Mapped points = Mapped points=sqrt(d)*u T
18 Roadmap I We need feature maps Explicit (feature maps) Implicit (kernel functions) Several algorithms need the inner products of features only! It is much easier to use implicit feature maps (kernels) Is it a kernel function??? 18
19 Roadmap II Is it a kernel function??? SVD, eigenvectors, eigenvalues Positive semi def. matrices Finite dim feature space Mercer s theorem, We have to think about the test data as well... eigenfunctions, eigenvalues Positive semi def. integral operators Infinite dim feature space (l ) If the kernel is pos. semi def., feature map construction 19
20 Mercer s theorem (*) 1 variable 0
21 Mercer s theorem... 1
22 Roadmap III We want to know which functions are kernels How to make new kernels from old kernels? The polynomial kernel: We will show another way using RKHS: Inner product=???
23 Ready for the details? ;)
24 Hard 1-dimensional Dataset What would SVMs do with this data? Not a big surprise x=0 Positive plane Negative plane Doesn t look like slack variables will save us this time taken from Andrew W. Moore 4
25 Hard 1-dimensional Dataset Make up a new feature! Sort of computed from original feature(s) z k ( x k, x k ) Separable! MAGIC! x=0 New features are sometimes called basis functions. Now drop this augmented data into our linear SVM. taken from Andrew W. Moore 5
26 Hard -dimensional Dataset O X X O Let us map this point to the 3 rd dimension... 6
27 Kernels and Linear Classifiers We will use linear classifiers in this feature space. 7
28 Picture is taken from R. Herbrich 8
29 Picture is taken from R. Herbrich 9
30 Kernels and Linear Classifiers Feature functions 30
31 Back to the Perceptron Example 31
32 The Perceptron The primal algorithm in the feature space 3
33 The primal algorithm in the feature space Picture is taken from R. Herbrich 33
34 The Perceptron 34
35 The Perceptron The Dual Algorithm in the feature space 35
36 The Dual Algorithm in the feature space Picture is taken from R. Herbrich 36
37 The Dual Algorithm in the feature space 37
38 Kernels Definition: (kernel) 38
39 Kernels Definition: (Gram matrix, kernel matrix) Definition: (Feature space, kernel space) 39
40 Kernel technique Definition: Lemma: The Gram matrix is symmetric, PSD matrix. Proof: 40
41 Kernel technique Key idea: 41
42 Kernel technique 4
43 Finite example r r = n n r 43
44 Finite example Lemma: Proof: 44
45 Kernel technique, Finite example We have seen: Lemma: These conditions are necessary 45
46 Kernel technique, Finite example Proof:... wrong in the Herbrich s book... 46
47 Kernel technique, Finite example Summary: How to generalize this to general sets??? 47
48 Integral operators, eigenfunctions Definition: Integral operator with kernel k(.,.) Remark: 48
49 From Vector domain to Functions Observe that each vector v = (v[1], v[],..., v[n]) is a mapping from the integers {1,,..., n} to < We can generalize this easily to INFINITE domain w = (w[1], w[],..., w[n],...) where w is mapping from {1,,...} to < 1 i 1 j 1 G v 1 49
50 From Vector domain to Functions From integers we can further extend to < or < m Strings Graphs Sets Whatever 50
51 L p and l p spaces. Picture is taken from R. Herbrich 51
52 L p and l p spaces Picture is taken from R. Herbrich 5
53 L and l special cases Picture is taken from R. Herbrich 53
54 Kernels Definition: inner product, Hilbert spaces 54
55 Integral operators, eigenfunctions Definition: Eigenvalue, Eigenfunction 55
56 Positive (semi) definite operators Definition: Positive Definite Operator 56
57 Mercer s theorem (*) 1 variable 57
58 Mercer s theorem... 58
59 A nicer characterization Theorem: nicer kernel characterization 59
60 Kernel Families Kernels have the intuitive meaning of similarity measure between objects. So far we have seen two ways for making a linear classifier nonlinear in the input space: 1. (explicit) Choosing a mapping ) Mercer kernel k. (implicit) Choosing a Mercer kernel k ) Mercer map 60
61 Designing new kernels from kernels are also kernels. Picture is taken from R. Herbrich 61
62 Designing new kernels from kernels Picture is taken from R. Herbrich 6
63 Designing new kernels from kernels 63
64 Note: Kernels on inner product spaces 64
65 Picture is taken from R. Herbrich 65
66 Polynomials of degree d Common Kernels Polynomials of degree up to d Sigmoid Gaussian kernels Equivalent to (x) of infinite dimensionality! 66
67 Note: The RBF kernel Proof: 67
68 Note: The RBF kernel Proof: Note: 68
69 The Polynomial kernel 69
70 Reminder: Hard 1-dimensional Dataset Make up a new feature! Sort of computed from original feature(s) z k ( x k, x k ) Separable! MAGIC! x=0 New features are sometimes called basis functions. Now drop this augmented data into our linear SVM. taken from Andrew W. Moore 70
71 New Features from Old Here: mapped by : x [x, x ] Found extra dimensions linearly separable! In general, Start with vector x N Want to add in x 1, x, Probably want other terms eg x x 7, Which ones to include? Why not ALL OF THEM??? 71
72 Special Case x=(x 1, x, x 3 ) (1, x 1, x, x 3, x 1, x, x 3, x 1 x, x 1 x 3, x x 3 ) 3 10, N=3, n=10; In general, the dimension of the quadratic map: N 1 N N N ( N )( N 1) N taken from Andrew W. Moore 7
73 73 Quadratic Basis Functions N N N N N N x x x x x x x x x x x x x x x x x x x : : : : : 1 ) ( Constant Term Linear Terms Pure Quadratic Terms Quadratic Cross-Terms What about those?? stay tuned Let taken from Andrew W. Moore
74 74 Quadratic Dot Products N N N N N N N N N N N b b b b b b b b b b b b b b b b b b an a a a a a a a a a a a a a a a a a : : : : : 1 : : : : : 1 ) ( ), ( b a 1 N i a i b i 1 N i a i b i 1 N i N i j j i j i b b a a taken from Andrew W. Moore
75 75 Quadratic Dot Products ) ( ), ( b a N i N i j j i j i N i i i N i i i b b a a b a a b ) ( 1 Now consider another fn of a and b (a b1) (a b) a b N i i i N i i i b a a b N i i i N i N j j j i i b a b a b a 1 ) ( N i i i N i N i j j j i i N i i i b a b a b a a b They re the same! And this is only O(N) to compute not O(N ) taken from Andrew W. Moore
76 Higher Order Polynomials Q kl y k y l (x k x l ) Polynomial (x) Quadratic All mn / terms up to degree Cubic All N 3 /6 terms up to degree 3 Quartic All N 4 /4 terms up to degree 4 Cost to Cost if 100 (a) (b) Cost to Cost if build Q kl inputs N=100 dim build Q kl 100 matrix: inputs matrix: inputs dim traditional sneaky inputs mn mr /4 500 Rm (a b+1) m N mr / 50 Rm N 3 m / m (a b+1) 3 N m / 50 m N 4 m / m (a b+1) 4 N m / 50 m taken from Andrew W. Moore 76
77 The Polynomial kernel, General case We are going to map these to a larger space We want to show that this k is a kernel function 77
78 The Polynomial kernel, General case We are going to map these to a larger space P factors 78
79 The Polynomial kernel, General case We already know: We want to get k in this form: 79
80 We already know: The Polynomial kernel For example 80
81 The Polynomial kernel 81
82 The Polynomial kernel ) k is really a kernel! 8
83 Reproducing Kernel Hilbert Spaces 83
84 1., RKHS, Motivation., Now, we show another way using RKHS What objective do we want to optimize? 84
85 RKHS, Motivation 3., How can we minimize the objective over functions??? Be PARAMETRIC!!!... (nope, we do not like that...) Use RKHS, and suddenly the problem will be finite dimensional optimization only (yummy...) The Representer theorem will help us here 1 st term, empirical loss nd term, regularization 85
86 Reproducing Kernel Hilbert Spaces Now, we show another way using RKHS Completing (closing) a pre-hilbert space ) Hilbert space 86
87 Reproducing Kernel Hilbert Spaces The inner product: (*) 87
88 Reproducing Kernel Hilbert Spaces Note: Proof: (*) 88
89 Reproducing Kernel Hilbert Spaces Lemma: Pre-Hilbert space: Like the Euclidean space with rational scalars only Hilbert space: Like the Euclidean space with real scalars Proof: 89
90 Reproducing Kernel Hilbert Spaces Lemma: (Reproducing property) Lemma: The constructed features match to k Huhh... 90
91 Reproducing Kernel Hilbert Spaces Proof of property 4.,: rep. property CBS For CBS we don t need 4., we need only that <0,0>=0! 91
92 Methods to Construct Feature Spaces We now have two methods to construct feature maps from kernels Well, these feature spaces are all isomorph with each other... 9
93 The Representer Theorem In the perceptron problem we could use the dual algorithm, because we had this representation: 93
94 Theorem: The Representer Theorem 1 st term, empirical loss nd term, regularization 94
95 The Representer Theorem Message: Optimizing in general function classes is difficult, but in RKHS it is only finite! (m) dimensional problem Proof of Representer Theorem: 95
96 Proof of the Representer Theorem Proof of Representer Theorem 1 st term, empirical loss nd term, regularization 96
97 Proof of the Representer Theorem 1 st term, empirical loss nd term, regularization 97
98 Supervised Learning SVM using kernels Gaussian Processes Regression Classification Heteroscedastic case Later will come Unsupervised Learning Kernel Principal Component Analysis Kernel Independent Component Analysis Kernel Mutual Information Kernel Generalized Variance Kernel Canonical Correlation Analysis 98
99 If we still have time Automatic Relevance Machines Bayes Point Machines Kernels on other objects Kernels on graphs Kernels on strings Fisher kernels ANOVA kernels Learning kernels 99
100 Thanks for the Attention! 100
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