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CS 59000 Statistical Machine learning Lecture 13 Yuan (Alan) Qi Purdue CS Oct. 8 2008

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Outline Review of kernel trick, kernel ridge regression and kernel Principle Component Analysis Gaussian processes (GPs) From linear regression to GP GP for regression

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Kernel Trick 1. Reformulate an algorithm such that input vector enters only in the form of inner product. 2. Replace input x by its feature mapping: 3. Replace the inner product by a Kernel function: Examples: Kernel PCA, Kernel Fisher discriminant, Support Vector Machines

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Dual variables: Dual Representation for Ridge Regression

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Kernel Ridge Regression Using kernel trick: Minimize over dual variables:

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Generate Kernel Matrix Positive semidefinite Consider Gaussian kernel:

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Principle Component Analysis (PCA) Assume We have is a normalized eigenvector:

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Feature Mapping Eigen-problem in feature space

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Dual Variables Suppose, we have

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Eigen-problem in Feature Space (1)

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Eigen-problem in Feature Space (2) Normalization condition: Projection coefficient:

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General Case for Non-zero Mean Case Kernel Matrix:

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Gaussian Processes How kernels arise naturally in a Bayesian setting? Instead of assigning a prior on parameters w, we assign a prior on function value y. Infinite space in theory Finite space in practice (finite number of training set and test set)

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Linear Regression Revisited Let We have

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From Prior on Parameter to Prior on Function The prior on function value:

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Stochastic Process A stochastic process is specified by giving the joint distribution for any finite set of values in a consistent manner (Loosely speaking, it means that a marginalized joint distribution is the same as the joint distribution that is defined in the subspace.)

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Gaussian Processes The joint distribution of any variables is a multivariable Gaussian distribution. Without any prior knowledge, we often set mean to be 0. Then the GP is specified by the covariance :

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Impact of Kernel Function Covariance matrix : kernel function Application economics & finance

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Gaussian Process for Regression Likelihood: Prior: Marginal distribution:

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Samples of GP Prior over Functions

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Samples of Data Points

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Predictive Distribution is a Gaussian distribution with mean and variance:

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Predictive Mean We see the same form as kernel ridge regression and kernel PCA.

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GP Regression Discussion: the difference between GP regression and Bayesian regression with Gaussian basis functions?

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Marginal Distribution of Target Values

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