Statistical Methods for Data Mining
|
|
- Berniece Atkinson
- 5 years ago
- Views:
Transcription
1 Statistical Methods for Data Mining Kuangnan Fang Xiamen University
2 Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find a plane that separates the classes in feature space. If we cannot, we get creative in two ways: We soften what we mean by separates, and We enrich and enlarge the feature space so that separation is possible. 1/21
3 What is a Hyperplane? A hyperplane in p dimensions is a flat a dimension p 1. ne subspace of In general the equation for a hyperplane has the form X X p X p =0 In p = 2 dimensions a hyperplane is a line. If 0 = 0, the hyperplane goes through the origin, otherwise not. The vector =( 1, 2,, p) is called the normal vector it points in a direction orthogonal to the surface of a hyperplane. 2/21
4 Hyperplane in 2 Dimensions X β 1 X 1 +β 2 X 2 6= 4 β 1 X 1 +β 2 X 2 6=0 β=(β 1,β 2 ) β 1 X 1 +β 2 X 2 6=1.6 β 1 = 0.8 β 2 = X 1 3/21
5 hyperplane divid p-dimensional space into two halves
6 A hyperplane in two-dimensional space
7 Classification Using a Separating Hyperplane
8 Separating Hyperplanes X X X X 1 If f(x) = X p X p,thenf(x) > 0 for points on one side of the hyperplane, and f(x) < 0 for points on the other. If we code the colored points as Y i = +1 for blue, say, and Y i = 1 for mauve, then if Y i f(x i ) > 0 for all i, f(x) =0 defines a separating hyperplane. 4/21
9 Maximal Margin Classifier Among all separating hyperplanes, find the one that makes the biggest gap or margin between the two classes. X Constrained optimization problem maximize 0, 1,..., p M subject to px j=1 2 j =1, y i ( x i p x ip ) M for all i =1,...,N X1 5/21
10 Maximal Margin Classifier Among all separating hyperplanes, find the one that makes the biggest gap or margin between the two classes. X Constrained optimization problem maximize 0, 1,..., p M subject to px j=1 2 j =1, y i ( x i p x ip ) M for all i =1,...,N X1 This can be rephrased as a convex quadratic program, and solved e ciently. The function svm() in package e1071 solves this problem e ciently 5/21
11 Maximal margin classifier
12 Maximal margin classifier
13 Maximal margin classifier
14 Constrained Optimization
15 Lagrange Multiplier
16 Lagrange Multiplier
17 Lagrange Multiplier
18 Lagrange Multiplier
19 Quadratic Programming
20 Quadratic Programming
21 Quadratic Programming
22 Quadratic Programming
23 Support Vectors
24 Non-separable Data X The data on the left are not separable by a linear boundary. This is often the case, unless N<p X1 6/21
25 Noisy Data X X X X 1 Sometimes the data are separable, but noisy. This can lead to a poor solution for the maximal-margin classifier. 7/21
26 Noisy Data X X X X 1 Sometimes the data are separable, but noisy. This can lead to a poor solution for the maximal-margin classifier. The support vector classifier maximizes a soft margin. 7/21
27 Support Vector Classifier X X X X 1 maximize M subject to 0, 1,..., p, 1,..., n px j=1 2 j =1, y i ( x i1 + 2 x i p x ip ) M(1 i ), nx i 0, i apple C, i=1 8/21
28 Maximal margin classifier
29 C is a regularization parameter X X1 X X X2 X X X1 9/21
30 Support vector classifier we can drop the norm constraint on, define M = 1 the equivalent form This is the usual way the support vector classifier is defined for the non- separable case. points well inside their class boundary do not play a big role in shaping the boundary.
31 Computing the Support Vector Classifier
32 Computing the Support Vector Classifier
33 Computing the Support Vector Classifier
34 Computing the Support Vector Classifier
35 The SVM as a Penalization Method The standard SVM: min 0,, 1 2 k P k2 + C n i i=1 subject to i 0,y i ( 0 + x > i ) 1 i,i=1, 2,,n. (1) where =( 1, 2,, n ) are the slack variables and C>0 is a constant. The constraint of (1) can be written as Then Objective function: 1 2 k k2 + C i [1 y i ( 0 + x > i )] +, i =1, 2,,n nx i=1 i 1 2 k k2 + C nx [1 y i ( 0 + x > i )] + If and only if i =[1 y i ( 0 + x > i )] +, the above inequality takes =. i=1
36 The SVM as a Penalization Method Then the solution of 0, in (1) is the same as the solution of the following optimization problem : min 0, 1 2 k k2 + C nx [1 y i ( 0 + x > i )] + i=1 Let =1/C,The above optimization problem is equivalent to min 0, nx [1 y i ( 0 + x > i )] + + k k 2. (2) 2 i=1 This has the form loss + penalty, which is a familiar paradigm in function estimation.
37 Implication of the Hinge Loss Consider the expectation of the loss function: E(`(yf) x) =`(f) Pr(y =1 x) +`( f) Pr(y = 1 x) Minimize expected loss when `(.) is hinge loss or 0-1 loss, obtain: f(x) =sign Pr(y =1 x) loss is the true loss function in binary classification. But the following optimization problem is di cult. 1 np min I [yi f(x f n i )<0] i=1 Since the solution to minimize the expected loss is the same, the hinge loss is also called the surrogate loss function. Compared with the perceptron loss, the hinge loss is not only classified correctly, but also the loss is only 0 when the confidence is high enough, that is, the hinge loss has higher requirements for learning.
38 Compare with other commonly used loss functions SVM Hinge Loss: [1 yf(x)] + Logistic Regression Loss: log[1 + e yf(x) ] Squared Error: [y f(x)] 2 =[1 yf(x)] 2 Huberized 8 Hinge Loss: < 0, yf > 1 `(yf) = (1 yf) 2 /(2 ), 1 <yfapple 1 : 1 t /2, yf apple 1. where >0 is a pre-specfied constant. Examination of the hinge loss function shows that it is reasonable for two-class classification, when compared to Logistic loss and Squared error. Logistic loss has similar tails as the SVM loss, but there is never zero penalty for points well inside their margin. Squared error gives a quadratic penalty, and points well inside their own margin have a strong influence on the model as well. Rosset and Zhu (2007) proposed a Huberized hinge loss, whichisvery similar to hinge loss in shape.
39 Hinge Loss and Huberized Loss As the decreases, the hubrized loss gets closer to the hinge loss. In fact, when! 0, the limit of hubrized loss is the hinge loss. Unlike the hinge loss, the huberized loss function is di erentiable everywhere. It is easy to prove that the huberized loss is satisfied arg min E[`(yf) x] =2Pr(y =1 x) f E(y x) > 0, sign[pr(y =1 x) 1 2 ] > 0. 1=E(y x) Rosset and Ji Zhu(2007): In the model with logistic loss, huberized loss, and square hinge loss with `1-penalty, the impact of outliers on the received huberized loss model is minimal.
40 `1-SVM The penalty of SVM is called the ridge penalty. It is well known that this shrinkage has the e ect of controlling the variances of ˆ; Hence possibly improves the fitted models prediction accuracy, especially when there are many highly correlated features. The SVM uses all variables in classification, which could be a great disadvantage in high-dimensional classification. Apply lasso penalty (`1-norm) to the SVM, Consider the optimization problem(`1-svm) 1 np min [1 y i ( 0 + x > i )] 0, n + + k k 1. i=1? the `1-SVM is able to automatically discard irrelevant features.? In the presence of many noise variables, the `1-SVM has significant advantages over the standard SVM.
41 `1-SVM The optimization problem of `1-SVM is a linear program with many constraints, and e cient algorithms can be complex. The solution paths can have many jumps, and show many discontinuities. For this reason, some authors prefer to replace the usual hinge loss with other smooth loss function, such as square hinge loss, huberized hinge loss, etc. (Zhu, Rosset, Hastie, 2004). The number of variables selected by the `1-norm penalty is upper bounded by the sample size n. For highly correlated and relevant variables, the `1-norm penalty tends to select only one or a few of them.
42 The hybrid huberized SVM(HHSVM) Li Wang, Ji Zhu and Hui Zou(2007) apply the elastic-net penalty to the SVM and propose the HHSVM: min ( 0, ) nx i=1 `(y i ( 0 + x > i )) + 1 k k k k2 2. (3) Where the loss function is huberized hinge loss: 8 < 0, yf > 1 `(yf) = (1 yf) 2 /(2 ), 1 <yfapple 1 : 1 yf /2, yf apple 1, and >0 is a pre-specified constant. The delfault choice for is 2 unless specified otherwise.
43 The hybrid huberized SVM(HHSVM) The advantage of Elastic-net penalty: 1. The elastic-net penalty retains the variable selection feature of the `1-norm penalty, and the number of selected variables is no longer bounded by n. 2. The elastic-net penalty tends to provide similar estimated coe cients for highly correlated variables. Theorem Let ˆ0 and ˆ denote the solution for problem (3). For any pair (j, j 0 ),wehave ˆj ˆj0 apple 1 kx j x j 0k = nx x ij x ij 0 i=1 If the input vector x j and x j 0 are centered and normalized, then ˆj ˆj0 apple 1 p n p kx j x j 0k = 2(1 ) 2 2 Where is the sample correlation between x j and x j 0.
44 The hybrid huberized SVM(HHSVM) GCD algorithm(yi Yang and Hui Zou(2014)) Define r i = y i ( 0 + x > i ), p 1, 2 ( j)= 1 j j, F ( j 0, ) = 1 n nx `(r i + y i x ij ( j j)) + p 1, ( 2 j) (4) i=1 Approximate the F function in (4) by a penalized quadratic function defined as Q( j 0, ) = 1 n nx `(r i )+ 1 n i=1 nx `0(r i )y i x ij ( j i=1 + 1 ( j j) 2 + p 1, 2 ( j) j) (5)
45 The hybrid huberized SVM(HHSVM) GCD algorithm(yi Yang and Hui Zou(2014)) By a simple soft-thresholding rule, can easily solve the minimizer of (5): np S( 2 1 j ˆC j = arg min Q( j 0, ) n `0(r i )y i x ij, 1) i=1 (6) =. 2/ + 2 j where S(z,t) =( z t) + sgn(z). Similarly, consider minimizing a quadratic approximation Q( 0 0, ) = 1 nx `(r i )+ 1 nx `0(r i )y i ( 0 0)+ 1 ( 0 0) 2. n n i=1 which has a minimizer ˆC 0 i=1 = arg min Q( 0 0, ) = np `0(r i )y i. n i=1 (7) (8)
46 The hybrid huberized SVM(HHSVM) Algorithm: The GCD algorithm for the HHSVM. 1. Initialize ( 0, ). 2. Iterate (a)(b) until convergence (a) Cyclic coordinate descent: for j =1, 2,,p. (a.1) Compute r i = y i( 0 + x > ). i (a.2) Compute r i = y i( 0 + x > ). i S( 2 j ˆCj = 1 n (a.3) j = ˆC j, j = j +1. (b) Update the intercept term (b.1) Compute r i = y i( 0 + x > i (b.2) Compute (b.3) 0 = ˆC 0. ˆC0 = 0 np `0(r i)y ix ij, 1) i=1. 2/ ). np `0(r i)y i. n i=1
47 Linear boundary can fail X Sometime a linear boundary simply won t work, no matter what value of C. The example on the left is such a case. What to do? X 1 10 / 21
48 Feature Expansion Enlarge the space of features by including transformations; e.g. X1 2, X3 1, X 1X 2, X 1 X2 2,... Hence go from a p-dimensional space to a M>pdimensional space. Fit a support-vector classifier in the enlarged space. This results in non-linear decision boundaries in the original space. 11 / 21
49 Feature Expansion Enlarge the space of features by including transformations; e.g. X1 2, X3 1, X 1X 2, X 1 X2 2,... Hence go from a p-dimensional space to a M>pdimensional space. Fit a support-vector classifier in the enlarged space. This results in non-linear decision boundaries in the original space. Example: Suppose we use (X 1,X 2,X 2 1,X2 2,X 1X 2 ) instead of just (X 1,X 2 ). Then the decision boundary would be of the form X X X X X 1 X 2 =0 This leads to nonlinear decision boundaries in the original space (quadratic conic sections). 11 / 21
50 hyperplane divid p-dimensional space into two halves
51 Here we use a basis expansion of cubic polynomials From 2 variables to 9 The support-vector classifier in the enlarged space solves the problem in the lower-dimensional space Cubic Polynomials X X X 1 12 / 21
52 Here we use a basis expansion of cubic polynomials From 2 variables to 9 The support-vector classifier in the enlarged space solves the problem in the lower-dimensional space Cubic Polynomials X X X X X X X X 1 X X X X 1 X X 2 1 X 2 =0 12 / 21
53 Nonlinearities and Kernels Polynomials (especially high-dimensional ones) get wild rather fast. There is a more elegant and controlled way to introduce nonlinearities in support-vector classifiers through the use of kernels. Before we discuss these, we must understand the role of inner products in support-vector classifiers. 13 / 21
54 Inner products and support vectors px hx i,x i 0i = x ij x i 0 j j=1 inner product between vectors 14 / 21
55 Inner products and support vectors px hx i,x i 0i = x ij x i 0 j j=1 inner product between vectors The linear support vector classifier can be represented as nx f(x) = 0 + i hx, x i i n parameters i=1 14 / 21
56 Inner products and support vectors px hx i,x i 0i = x ij x i 0 j j=1 inner product between vectors The linear support vector classifier can be represented as nx f(x) = 0 + i hx, x i i n parameters i=1 To estimate the parameters 1,..., n and 0, all we need are the n 2 inner products hx i,x i 0i between all pairs of training observations. 14 / 21
57 Inner products and support vectors px hx i,x i 0i = x ij x i 0 j j=1 inner product between vectors The linear support vector classifier can be represented as nx f(x) = 0 + i hx, x i i n parameters i=1 To estimate the parameters 1,..., n and 0, all we need are the n 2 inner products hx i,x i 0i between all pairs of training observations. It turns out that most of the ˆ i can be zero: f(x) = 0 + X i2s ˆ i hx, x i i S is the support set of indices i such that ˆ i > 0. [see slide 8] 14 / 21
58 Kernels and Support Vector Machines If we can compute inner-products between observations, we can fit a SV classifier. Can be quite abstract! 15 / 21
59 Kernels and Support Vector Machines If we can compute inner-products between observations, we can fit a SV classifier. Can be quite abstract! Some special kernel functions can do this for us. E.g. 0 K(x i,x i 1 px x ij x i A 0 j j=1 computes the inner-products needed for d dimensional polynomials basis functions! p+d d d 15 / 21
60 Kernels and Support Vector Machines If we can compute inner-products between observations, we can fit a SV classifier. Can be quite abstract! Some special kernel functions can do this for us. E.g. 0 K(x i,x i 1 px x ij x i A 0 j j=1 computes the inner-products needed for d dimensional polynomials p+d d basis functions! Try it for p =2and d =2. d 15 / 21
61 Kernels and Support Vector Machines If we can compute inner-products between observations, we can fit a SV classifier. Can be quite abstract! Some special kernel functions can do this for us. E.g. 0 K(x i,x i 1 px x ij x i A 0 j j=1 computes the inner-products needed for d dimensional polynomials p+d d basis functions! Try it for p =2and d =2. The solution has the form d f(x) = 0 + X i2s ˆ i K(x, x i ). 15 / 21
62 Radial Kernel K(x i,x i 0)=exp( px (x ij x i 0 j) 2 ). j=1 X f(x) = 0 + X i2s ˆ i K(x, x i ) Implicit feature space; very high dimensional. Controls variance by squashing down most dimensions severely X 1 16 / 21
63 Example: Heart Data True positive rate Support Vector Classifier LDA True positive rate Support Vector Classifier SVM: γ=10 3 SVM: γ=10 2 SVM: γ= False positive rate False positive rate ROC curve is obtained by changing the threshold 0 to threshold t in ˆf(X) >t, and recording false positive and true positive rates as t varies. Here we see ROC curves on training data. 17 / 21
64 ROC curve
65 ROC
66 ROC curve
67 SVMs: more than 2 classes? The SVM as defined works for K = 2 classes. What do we do if we have K>2 classes? 19 / 21
68 SVMs: more than 2 classes? The SVM as defined works for K = 2 classes. What do we do if we have K>2 classes? OVA One versus All. Fit K di erent 2-class SVM classifiers ˆf k (x), k=1,...,k; each class versus the rest. Classify x to the class for which ˆf k (x ) is largest. 19 / 21
69 SVMs: more than 2 classes? The SVM as defined works for K = 2 classes. What do we do if we have K>2 classes? OVA One versus All. Fit K di erent 2-class SVM classifiers ˆf k (x), k=1,...,k; each class versus the rest. Classify x to the class for which ˆf k (x ) is largest. K OVO One versus One. Fit all 2 pairwise classifiers ˆf k`(x). Classify x to the class that wins the most pairwise competitions. 19 / 21
70 SVMs: more than 2 classes? The SVM as defined works for K = 2 classes. What do we do if we have K>2 classes? OVA One versus All. Fit K di erent 2-class SVM classifiers ˆf k (x), k=1,...,k; each class versus the rest. Classify x to the class for which ˆf k (x ) is largest. K OVO One versus One. Fit all 2 pairwise classifiers ˆf k`(x). Classify x to the class that wins the most pairwise competitions. Which to choose? If K is not too large, use OVO. 19 / 21
71 Support Vector versus Logistic Regression? With f(x) = X p X p can rephrase support-vector classifier optimization as 8 < nx minimize max [0, 1 y 0, 1,..., p : i f(x i )] + i=1 px j=1 2 j 9 = ; Loss SVM Loss Logistic Regression Loss This has the form loss plus penalty. The loss is known as the hinge loss. Very similar to loss in logistic regression (negative log-likelihood) y i( 0 + 1x i px ip) 20 / 21
72 Which to use: SVM or Logistic Regression When classes are (nearly) separable, SVM does better than LR. So does LDA. When not, LR (with ridge penalty) and SVM very similar. If you wish to estimate probabilities, LR is the choice. For nonlinear boundaries, kernel SVMs are popular. Can use kernels with LR and LDA as well, but computations are more expensive. 21 / 21
73 Support vector regression
74 Support vector regression
75 Support vector regression
76 Support vector regression
77 Support vector regression
78 Support vector regression
Statistical Methods for SVM
Statistical Methods for SVM Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find a plane that separates the classes in feature space. If we cannot,
More informationSupport Vector Machines
Support Vector Machines Here we approach the two-class classification problem in a direct way: We try and find a plane that separates the classes in feature space. If we cannot, we get creative in two
More informationMachine Learning Practice Page 2 of 2 10/28/13
Machine Learning 10-701 Practice Page 2 of 2 10/28/13 1. True or False Please give an explanation for your answer, this is worth 1 pt/question. (a) (2 points) No classifier can do better than a naive Bayes
More informationL5 Support Vector Classification
L5 Support Vector Classification Support Vector Machine Problem definition Geometrical picture Optimization problem Optimization Problem Hard margin Convexity Dual problem Soft margin problem Alexander
More informationSupport Vector Machine (SVM) and Kernel Methods
Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2014 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin
More informationLinear & nonlinear classifiers
Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1394 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1394 1 / 34 Table
More informationSupport Vector Machines for Classification: A Statistical Portrait
Support Vector Machines for Classification: A Statistical Portrait Yoonkyung Lee Department of Statistics The Ohio State University May 27, 2011 The Spring Conference of Korean Statistical Society KAIST,
More informationSVMs: Non-Separable Data, Convex Surrogate Loss, Multi-Class Classification, Kernels
SVMs: Non-Separable Data, Convex Surrogate Loss, Multi-Class Classification, Kernels Karl Stratos June 21, 2018 1 / 33 Tangent: Some Loose Ends in Logistic Regression Polynomial feature expansion in logistic
More informationFinal Overview. Introduction to ML. Marek Petrik 4/25/2017
Final Overview Introduction to ML Marek Petrik 4/25/2017 This Course: Introduction to Machine Learning Build a foundation for practice and research in ML Basic machine learning concepts: max likelihood,
More informationSupport Vector Machines
Wien, June, 2010 Paul Hofmarcher, Stefan Theussl, WU Wien Hofmarcher/Theussl SVM 1/21 Linear Separable Separating Hyperplanes Non-Linear Separable Soft-Margin Hyperplanes Hofmarcher/Theussl SVM 2/21 (SVM)
More informationSupport Vector Machine (SVM) and Kernel Methods
Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2015 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin
More informationSupport Vector Machines and Kernel Methods
2018 CS420 Machine Learning, Lecture 3 Hangout from Prof. Andrew Ng. http://cs229.stanford.edu/notes/cs229-notes3.pdf Support Vector Machines and Kernel Methods Weinan Zhang Shanghai Jiao Tong University
More informationCS6375: Machine Learning Gautam Kunapuli. Support Vector Machines
Gautam Kunapuli Example: Text Categorization Example: Develop a model to classify news stories into various categories based on their content. sports politics Use the bag-of-words representation for this
More informationPattern Recognition 2018 Support Vector Machines
Pattern Recognition 2018 Support Vector Machines Ad Feelders Universiteit Utrecht Ad Feelders ( Universiteit Utrecht ) Pattern Recognition 1 / 48 Support Vector Machines Ad Feelders ( Universiteit Utrecht
More informationPattern Recognition and Machine Learning. Perceptrons and Support Vector machines
Pattern Recognition and Machine Learning James L. Crowley ENSIMAG 3 - MMIS Fall Semester 2016 Lessons 6 10 Jan 2017 Outline Perceptrons and Support Vector machines Notation... 2 Perceptrons... 3 History...3
More informationLinear & nonlinear classifiers
Linear & nonlinear classifiers Machine Learning Hamid Beigy Sharif University of Technology Fall 1396 Hamid Beigy (Sharif University of Technology) Linear & nonlinear classifiers Fall 1396 1 / 44 Table
More informationCOMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017
COMS 4721: Machine Learning for Data Science Lecture 10, 2/21/2017 Prof. John Paisley Department of Electrical Engineering & Data Science Institute Columbia University FEATURE EXPANSIONS FEATURE EXPANSIONS
More informationLecture 18: Kernels Risk and Loss Support Vector Regression. Aykut Erdem December 2016 Hacettepe University
Lecture 18: Kernels Risk and Loss Support Vector Regression Aykut Erdem December 2016 Hacettepe University Administrative We will have a make-up lecture on next Saturday December 24, 2016 Presentations
More informationSupport Vector Machines
Support Vector Machines Hypothesis Space variable size deterministic continuous parameters Learning Algorithm linear and quadratic programming eager batch SVMs combine three important ideas Apply optimization
More informationDeep Learning for Computer Vision
Deep Learning for Computer Vision Lecture 4: Curse of Dimensionality, High Dimensional Feature Spaces, Linear Classifiers, Linear Regression, Python, and Jupyter Notebooks Peter Belhumeur Computer Science
More informationSupport Vector Machine (continued)
Support Vector Machine continued) Overlapping class distribution: In practice the class-conditional distributions may overlap, so that the training data points are no longer linearly separable. We need
More informationLecture 10: A brief introduction to Support Vector Machine
Lecture 10: A brief introduction to Support Vector Machine Advanced Applied Multivariate Analysis STAT 2221, Fall 2013 Sungkyu Jung Department of Statistics, University of Pittsburgh Xingye Qiao Department
More informationIntroduction to Machine Learning
1, DATA11002 Introduction to Machine Learning Lecturer: Teemu Roos TAs: Ville Hyvönen and Janne Leppä-aho Department of Computer Science University of Helsinki (based in part on material by Patrik Hoyer
More informationStat542 (F11) Statistical Learning. First consider the scenario where the two classes of points are separable.
Linear SVM (separable case) First consider the scenario where the two classes of points are separable. It s desirable to have the width (called margin) between the two dashed lines to be large, i.e., have
More informationMachine Learning And Applications: Supervised Learning-SVM
Machine Learning And Applications: Supervised Learning-SVM Raphaël Bournhonesque École Normale Supérieure de Lyon, Lyon, France raphael.bournhonesque@ens-lyon.fr 1 Supervised vs unsupervised learning Machine
More informationSupport Vector Machine (SVM) and Kernel Methods
Support Vector Machine (SVM) and Kernel Methods CE-717: Machine Learning Sharif University of Technology Fall 2016 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin
More informationNon-Bayesian Classifiers Part II: Linear Discriminants and Support Vector Machines
Non-Bayesian Classifiers Part II: Linear Discriminants and Support Vector Machines Selim Aksoy Department of Computer Engineering Bilkent University saksoy@cs.bilkent.edu.tr CS 551, Fall 2018 CS 551, Fall
More informationSupport Vector Machines. Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar
Data Mining Support Vector Machines Introduction to Data Mining, 2 nd Edition by Tan, Steinbach, Karpatne, Kumar 02/03/2018 Introduction to Data Mining 1 Support Vector Machines Find a linear hyperplane
More informationSupport Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology. M. Soleymani Fall 2012
Support Vector Machine (SVM) & Kernel CE-717: Machine Learning Sharif University of Technology M. Soleymani Fall 2012 Linear classifier Which classifier? x 2 x 1 2 Linear classifier Margin concept x 2
More informationStatistical Pattern Recognition
Statistical Pattern Recognition Support Vector Machine (SVM) Hamid R. Rabiee Hadi Asheri, Jafar Muhammadi, Nima Pourdamghani Spring 2013 http://ce.sharif.edu/courses/91-92/2/ce725-1/ Agenda Introduction
More informationReview: Support vector machines. Machine learning techniques and image analysis
Review: Support vector machines Review: Support vector machines Margin optimization min (w,w 0 ) 1 2 w 2 subject to y i (w 0 + w T x i ) 1 0, i = 1,..., n. Review: Support vector machines Margin optimization
More informationData Mining. Linear & nonlinear classifiers. Hamid Beigy. Sharif University of Technology. Fall 1396
Data Mining Linear & nonlinear classifiers Hamid Beigy Sharif University of Technology Fall 1396 Hamid Beigy (Sharif University of Technology) Data Mining Fall 1396 1 / 31 Table of contents 1 Introduction
More informationSupport Vector Machine I
Support Vector Machine I Jia-Bin Huang ECE-5424G / CS-5824 Virginia Tech Spring 2019 Administrative Please use piazza. No emails. HW 0 grades are back. Re-grade request for one week. HW 1 due soon. HW
More informationDiscriminative Models
No.5 Discriminative Models Hui Jiang Department of Electrical Engineering and Computer Science Lassonde School of Engineering York University, Toronto, Canada Outline Generative vs. Discriminative models
More informationMachine Learning. Support Vector Machines. Manfred Huber
Machine Learning Support Vector Machines Manfred Huber 2015 1 Support Vector Machines Both logistic regression and linear discriminant analysis learn a linear discriminant function to separate the data
More informationJeff Howbert Introduction to Machine Learning Winter
Classification / Regression Support Vector Machines Jeff Howbert Introduction to Machine Learning Winter 2012 1 Topics SVM classifiers for linearly separable classes SVM classifiers for non-linearly separable
More informationPerceptron Revisited: Linear Separators. Support Vector Machines
Support Vector Machines Perceptron Revisited: Linear Separators Binary classification can be viewed as the task of separating classes in feature space: w T x + b > 0 w T x + b = 0 w T x + b < 0 Department
More informationCIS 520: Machine Learning Oct 09, Kernel Methods
CIS 520: Machine Learning Oct 09, 207 Kernel Methods Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture They may or may not cover all the material discussed
More informationSupport Vector Machines
EE 17/7AT: Optimization Models in Engineering Section 11/1 - April 014 Support Vector Machines Lecturer: Arturo Fernandez Scribe: Arturo Fernandez 1 Support Vector Machines Revisited 1.1 Strictly) Separable
More informationMachine Learning Support Vector Machines. Prof. Matteo Matteucci
Machine Learning Support Vector Machines Prof. Matteo Matteucci Discriminative vs. Generative Approaches 2 o Generative approach: we derived the classifier from some generative hypothesis about the way
More informationLecture 14 : Online Learning, Stochastic Gradient Descent, Perceptron
CS446: Machine Learning, Fall 2017 Lecture 14 : Online Learning, Stochastic Gradient Descent, Perceptron Lecturer: Sanmi Koyejo Scribe: Ke Wang, Oct. 24th, 2017 Agenda Recap: SVM and Hinge loss, Representer
More informationLecture 9: Large Margin Classifiers. Linear Support Vector Machines
Lecture 9: Large Margin Classifiers. Linear Support Vector Machines Perceptrons Definition Perceptron learning rule Convergence Margin & max margin classifiers (Linear) support vector machines Formulation
More informationMulti-class SVMs. Lecture 17: Aykut Erdem April 2016 Hacettepe University
Multi-class SVMs Lecture 17: Aykut Erdem April 2016 Hacettepe University Administrative We will have a make-up lecture on Saturday April 23, 2016. Project progress reports are due April 21, 2016 2 days
More informationClassification and Support Vector Machine
Classification and Support Vector Machine Yiyong Feng and Daniel P. Palomar The Hong Kong University of Science and Technology (HKUST) ELEC 5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline
More informationDiscriminative Models
No.5 Discriminative Models Hui Jiang Department of Electrical Engineering and Computer Science Lassonde School of Engineering York University, Toronto, Canada Outline Generative vs. Discriminative models
More informationCSC 411 Lecture 17: Support Vector Machine
CSC 411 Lecture 17: Support Vector Machine Ethan Fetaya, James Lucas and Emad Andrews University of Toronto CSC411 Lec17 1 / 1 Today Max-margin classification SVM Hard SVM Duality Soft SVM CSC411 Lec17
More informationMark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation.
CS 189 Spring 2015 Introduction to Machine Learning Midterm You have 80 minutes for the exam. The exam is closed book, closed notes except your one-page crib sheet. No calculators or electronic items.
More informationNeural Networks. Prof. Dr. Rudolf Kruse. Computational Intelligence Group Faculty for Computer Science
Neural Networks Prof. Dr. Rudolf Kruse Computational Intelligence Group Faculty for Computer Science kruse@iws.cs.uni-magdeburg.de Rudolf Kruse Neural Networks 1 Supervised Learning / Support Vector Machines
More informationChapter 9. Support Vector Machine. Yongdai Kim Seoul National University
Chapter 9. Support Vector Machine Yongdai Kim Seoul National University 1. Introduction Support Vector Machine (SVM) is a classification method developed by Vapnik (1996). It is thought that SVM improved
More informationBANA 7046 Data Mining I Lecture 6. Other Data Mining Algorithms 1
BANA 7046 Data Mining I Lecture 6. Other Data Mining Algorithms 1 Shaobo Li University of Cincinnati 1 Partially based on Hastie, et al. (2009) ESL, and James, et al. (2013) ISLR Data Mining I Lecture
More informationSupport Vector Machines
Support Vector Machines Support vector machines (SVMs) are one of the central concepts in all of machine learning. They are simply a combination of two ideas: linear classification via maximum (or optimal
More informationIndirect Rule Learning: Support Vector Machines. Donglin Zeng, Department of Biostatistics, University of North Carolina
Indirect Rule Learning: Support Vector Machines Indirect learning: loss optimization It doesn t estimate the prediction rule f (x) directly, since most loss functions do not have explicit optimizers. Indirection
More information10/05/2016. Computational Methods for Data Analysis. Massimo Poesio SUPPORT VECTOR MACHINES. Support Vector Machines Linear classifiers
Computational Methods for Data Analysis Massimo Poesio SUPPORT VECTOR MACHINES Support Vector Machines Linear classifiers 1 Linear Classifiers denotes +1 denotes -1 w x + b>0 f(x,w,b) = sign(w x + b) How
More informationStatistical Data Mining and Machine Learning Hilary Term 2016
Statistical Data Mining and Machine Learning Hilary Term 2016 Dino Sejdinovic Department of Statistics Oxford Slides and other materials available at: http://www.stats.ox.ac.uk/~sejdinov/sdmml Naïve Bayes
More informationSUPPORT VECTOR MACHINE
SUPPORT VECTOR MACHINE Mainly based on https://nlp.stanford.edu/ir-book/pdf/15svm.pdf 1 Overview SVM is a huge topic Integration of MMDS, IIR, and Andrew Moore s slides here Our foci: Geometric intuition
More informationLINEAR CLASSIFIERS. J. Elder CSE 4404/5327 Introduction to Machine Learning and Pattern Recognition
LINEAR CLASSIFIERS Classification: Problem Statement 2 In regression, we are modeling the relationship between a continuous input variable x and a continuous target variable t. In classification, the input
More informationLinear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)
Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training
More informationIntroduction to SVM and RVM
Introduction to SVM and RVM Machine Learning Seminar HUS HVL UIB Yushu Li, UIB Overview Support vector machine SVM First introduced by Vapnik, et al. 1992 Several literature and wide applications Relevance
More informationSupport Vector Machine
Support Vector Machine Fabrice Rossi SAMM Université Paris 1 Panthéon Sorbonne 2018 Outline Linear Support Vector Machine Kernelized SVM Kernels 2 From ERM to RLM Empirical Risk Minimization in the binary
More informationConvex Optimization and Support Vector Machine
Convex Optimization and Support Vector Machine Problem 0. Consider a two-class classification problem. The training data is L n = {(x 1, t 1 ),..., (x n, t n )}, where each t i { 1, 1} and x i R p. We
More information9 Classification. 9.1 Linear Classifiers
9 Classification This topic returns to prediction. Unlike linear regression where we were predicting a numeric value, in this case we are predicting a class: winner or loser, yes or no, rich or poor, positive
More informationLinear Support Vector Machine. Classification. Linear SVM. Huiping Cao. Huiping Cao, Slide 1/26
Huiping Cao, Slide 1/26 Classification Linear SVM Huiping Cao linear hyperplane (decision boundary) that will separate the data Huiping Cao, Slide 2/26 Support Vector Machines rt Vector Find a linear Machines
More informationLinear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)
Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training
More informationRegularization Paths
December 2005 Trevor Hastie, Stanford Statistics 1 Regularization Paths Trevor Hastie Stanford University drawing on collaborations with Brad Efron, Saharon Rosset, Ji Zhu, Hui Zhou, Rob Tibshirani and
More informationSupport Vector Machines for Classification and Regression. 1 Linearly Separable Data: Hard Margin SVMs
E0 270 Machine Learning Lecture 5 (Jan 22, 203) Support Vector Machines for Classification and Regression Lecturer: Shivani Agarwal Disclaimer: These notes are a brief summary of the topics covered in
More informationIntroduction to Machine Learning. Regression. Computer Science, Tel-Aviv University,
1 Introduction to Machine Learning Regression Computer Science, Tel-Aviv University, 2013-14 Classification Input: X Real valued, vectors over real. Discrete values (0,1,2,...) Other structures (e.g.,
More informationSVAN 2016 Mini Course: Stochastic Convex Optimization Methods in Machine Learning
SVAN 2016 Mini Course: Stochastic Convex Optimization Methods in Machine Learning Mark Schmidt University of British Columbia, May 2016 www.cs.ubc.ca/~schmidtm/svan16 Some images from this lecture are
More informationEE613 Machine Learning for Engineers. Kernel methods Support Vector Machines. jean-marc odobez 2015
EE613 Machine Learning for Engineers Kernel methods Support Vector Machines jean-marc odobez 2015 overview Kernel methods introductions and main elements defining kernels Kernelization of k-nn, K-Means,
More informationKernel Machines. Pradeep Ravikumar Co-instructor: Manuela Veloso. Machine Learning
Kernel Machines Pradeep Ravikumar Co-instructor: Manuela Veloso Machine Learning 10-701 SVM linearly separable case n training points (x 1,, x n ) d features x j is a d-dimensional vector Primal problem:
More informationAnnouncements - Homework
Announcements - Homework Homework 1 is graded, please collect at end of lecture Homework 2 due today Homework 3 out soon (watch email) Ques 1 midterm review HW1 score distribution 40 HW1 total score 35
More informationWarm up: risk prediction with logistic regression
Warm up: risk prediction with logistic regression Boss gives you a bunch of data on loans defaulting or not: {(x i,y i )} n i= x i 2 R d, y i 2 {, } You model the data as: P (Y = y x, w) = + exp( yw T
More information11 More Regression; Newton s Method; ROC Curves
More Regression; Newton s Method; ROC Curves 59 11 More Regression; Newton s Method; ROC Curves LEAST-SQUARES POLYNOMIAL REGRESSION Replace each X i with feature vector e.g. (X i ) = [X 2 i1 X i1 X i2
More information> DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE GRAVIS 2016 BASEL. Logistic Regression. Pattern Recognition 2016 Sandro Schönborn University of Basel
Logistic Regression Pattern Recognition 2016 Sandro Schönborn University of Basel Two Worlds: Probabilistic & Algorithmic We have seen two conceptual approaches to classification: data class density estimation
More informationLINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES. Supervised Learning
LINEAR CLASSIFICATION, PERCEPTRON, LOGISTIC REGRESSION, SVC, NAÏVE BAYES Supervised Learning Linear vs non linear classifiers In K-NN we saw an example of a non-linear classifier: the decision boundary
More informationSupport Vector Machines for Classification and Regression
CIS 520: Machine Learning Oct 04, 207 Support Vector Machines for Classification and Regression Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may
More information18.9 SUPPORT VECTOR MACHINES
744 Chapter 8. Learning from Examples is the fact that each regression problem will be easier to solve, because it involves only the examples with nonzero weight the examples whose kernels overlap the
More informationLinear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)
Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers) Solution only depends on a small subset of training
More information(Kernels +) Support Vector Machines
(Kernels +) Support Vector Machines Machine Learning Torsten Möller Reading Chapter 5 of Machine Learning An Algorithmic Perspective by Marsland Chapter 6+7 of Pattern Recognition and Machine Learning
More informationUNIVERSITY of PENNSYLVANIA CIS 520: Machine Learning Final, Fall 2014
UNIVERSITY of PENNSYLVANIA CIS 520: Machine Learning Final, Fall 2014 Exam policy: This exam allows two one-page, two-sided cheat sheets (i.e. 4 sides); No other materials. Time: 2 hours. Be sure to write
More informationCS145: INTRODUCTION TO DATA MINING
CS145: INTRODUCTION TO DATA MINING 5: Vector Data: Support Vector Machine Instructor: Yizhou Sun yzsun@cs.ucla.edu October 18, 2017 Homework 1 Announcements Due end of the day of this Thursday (11:59pm)
More informationCOMP 875 Announcements
Announcements Tentative presentation order is out Announcements Tentative presentation order is out Remember: Monday before the week of the presentation you must send me the final paper list (for posting
More informationEXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING
EXAM IN STATISTICAL MACHINE LEARNING STATISTISK MASKININLÄRNING DATE AND TIME: August 30, 2018, 14.00 19.00 RESPONSIBLE TEACHER: Niklas Wahlström NUMBER OF PROBLEMS: 5 AIDING MATERIAL: Calculator, mathematical
More informationAn Improved 1-norm SVM for Simultaneous Classification and Variable Selection
An Improved 1-norm SVM for Simultaneous Classification and Variable Selection Hui Zou School of Statistics University of Minnesota Minneapolis, MN 55455 hzou@stat.umn.edu Abstract We propose a novel extension
More informationSupport Vector Machines
Support Vector Machines Le Song Machine Learning I CSE 6740, Fall 2013 Naïve Bayes classifier Still use Bayes decision rule for classification P y x = P x y P y P x But assume p x y = 1 is fully factorized
More informationMax Margin-Classifier
Max Margin-Classifier Oliver Schulte - CMPT 726 Bishop PRML Ch. 7 Outline Maximum Margin Criterion Math Maximizing the Margin Non-Separable Data Kernels and Non-linear Mappings Where does the maximization
More informationLearning by constraints and SVMs (2)
Statistical Techniques in Robotics (16-831, F12) Lecture#14 (Wednesday ctober 17) Learning by constraints and SVMs (2) Lecturer: Drew Bagnell Scribe: Albert Wu 1 1 Support Vector Ranking Machine pening
More informationApplied Machine Learning Annalisa Marsico
Applied Machine Learning Annalisa Marsico OWL RNA Bionformatics group Max Planck Institute for Molecular Genetics Free University of Berlin 22 April, SoSe 2015 Goals Feature Selection rather than Feature
More informationA Magiv CV Theory for Large-Margin Classifiers
A Magiv CV Theory for Large-Margin Classifiers Hui Zou School of Statistics, University of Minnesota June 30, 2018 Joint work with Boxiang Wang Outline 1 Background 2 Magic CV formula 3 Magic support vector
More informationEvaluation requires to define performance measures to be optimized
Evaluation Basic concepts Evaluation requires to define performance measures to be optimized Performance of learning algorithms cannot be evaluated on entire domain (generalization error) approximation
More informationCOMS 4771 Introduction to Machine Learning. James McInerney Adapted from slides by Nakul Verma
COMS 4771 Introduction to Machine Learning James McInerney Adapted from slides by Nakul Verma Announcements HW1: Please submit as a group Watch out for zero variance features (Q5) HW2 will be released
More informationNonlinear Support Vector Machines through Iterative Majorization and I-Splines
Nonlinear Support Vector Machines through Iterative Majorization and I-Splines P.J.F. Groenen G. Nalbantov J.C. Bioch July 9, 26 Econometric Institute Report EI 26-25 Abstract To minimize the primal support
More informationKernel Methods and Support Vector Machines
Kernel Methods and Support Vector Machines Oliver Schulte - CMPT 726 Bishop PRML Ch. 6 Support Vector Machines Defining Characteristics Like logistic regression, good for continuous input features, discrete
More informationLEAST ANGLE REGRESSION 469
LEAST ANGLE REGRESSION 469 Specifically for the Lasso, one alternative strategy for logistic regression is to use a quadratic approximation for the log-likelihood. Consider the Bayesian version of Lasso
More informationDATA MINING AND MACHINE LEARNING
DATA MINING AND MACHINE LEARNING Lecture 5: Regularization and loss functions Lecturer: Simone Scardapane Academic Year 2016/2017 Table of contents Loss functions Loss functions for regression problems
More informationMachine Learning and Data Mining. Support Vector Machines. Kalev Kask
Machine Learning and Data Mining Support Vector Machines Kalev Kask Linear classifiers Which decision boundary is better? Both have zero training error (perfect training accuracy) But, one of them seems
More informationLecture 10: Support Vector Machine and Large Margin Classifier
Lecture 10: Support Vector Machine and Large Margin Classifier Applied Multivariate Analysis Math 570, Fall 2014 Xingye Qiao Department of Mathematical Sciences Binghamton University E-mail: qiao@math.binghamton.edu
More informationSupport Vector Machines, Kernel SVM
Support Vector Machines, Kernel SVM Professor Ameet Talwalkar Professor Ameet Talwalkar CS260 Machine Learning Algorithms February 27, 2017 1 / 40 Outline 1 Administration 2 Review of last lecture 3 SVM
More informationMidterm Review CS 6375: Machine Learning. Vibhav Gogate The University of Texas at Dallas
Midterm Review CS 6375: Machine Learning Vibhav Gogate The University of Texas at Dallas Machine Learning Supervised Learning Unsupervised Learning Reinforcement Learning Parametric Y Continuous Non-parametric
More informationSupport Vector Machines.
Support Vector Machines www.cs.wisc.edu/~dpage 1 Goals for the lecture you should understand the following concepts the margin slack variables the linear support vector machine nonlinear SVMs the kernel
More informationSupport Vector Machine
Andrea Passerini passerini@disi.unitn.it Machine Learning Support vector machines In a nutshell Linear classifiers selecting hyperplane maximizing separation margin between classes (large margin classifiers)
More information