Machine learning continued Image source:

More about linear classifiers When the data is linearly separable, there may be more than one separator (hyperplane) Which separator is best?

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, 1998A Tutorial on Support Vector Machines for Pattern Recognition

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

Finding the maximum margin hyperplane 1.Maximize margin 2 / ||w|| 2.Correctly classify all training data: Quadratic optimization problem: C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998A Tutorial on Support Vector Machines for Pattern Recognition

Finding the maximum margin hyperplane Solution: C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998A Tutorial on Support Vector Machines for Pattern Recognition Support vector learned weight

Finding the maximum margin hyperplane Solution: b = y i – w·x i for any support vector Classification function (decision boundary): Notice that it relies on an inner product between the test point x and the support vectors x i Solving the optimization problem also involves computing the inner products x i · x j between all pairs of training points C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998A Tutorial on Support Vector Machines for Pattern Recognition

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: 0x 0 x 0 x x2x2 Nonlinear SVMs Slide credit: Andrew Moore

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

Nonlinear SVMs The kernel trick: instead of explicitly computing the lifting transformation φ(x), define a kernel function K such that K(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, 1998A Tutorial on Support Vector Machines for Pattern Recognition

Nonlinear kernel: Example Consider the mapping x2x2

Polynomial kernel:

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?

Gaussian kernel SV’s

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

SVMs: Pros and cons Pros Many publicly available SVM packages: 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 (esp. for nonlinear SVMs) –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

Beyond simple classification: Structured prediction Image Word Source: B. Taskar

Structured Prediction Sentence Parse tree Source: B. Taskar

Structured Prediction Sentence in two languages Word alignment Source: B. Taskar

Structured Prediction Amino-acid sequence Bond structure Source: B. Taskar

Structured Prediction Many image-based inference tasks can loosely be thought of as “structured prediction” Source: D. Ramanan model

Unsupervised Learning Idea: Given only unlabeled data as input, learn some sort of structure The objective is often more vague or subjective than in supervised learning This is more of an exploratory/descriptive data analysis

Unsupervised Learning Clustering –Discover groups of “similar” data points

Unsupervised Learning Quantization –Map a continuous input to a discrete (more compact) output 1 2 3

Dimensionality reduction, manifold learning –Discover a lower-dimensional surface on which the data lives Unsupervised Learning

Density estimation –Find a function that approximates the probability density of the data (i.e., value of the function is high for “typical” points and low for “atypical” points) –Can be used for anomaly detection

Semi-supervised learning Lots of data is available, but only small portion is labeled (e.g. since labeling is expensive) –Why is learning from labeled and unlabeled data better than learning from labeled data alone? ?

Active learning The learning algorithm can choose its own training examples, or ask a “teacher” for an answer on selected inputs S. Vijayanarasimhan and K. Grauman, “Cost-Sensitive Active Visual Category Learning,” 2009

Lifelong learning

Lifelong learning

Xinlei Chen, Abhinav Shrivastava and Abhinav Gupta. NEIL: Extracting Visual Knowledge from Web Data. In ICCV 2013NEIL: Extracting Visual Knowledge from Web Data