CS Statistical Machine learning Lecture 13 Yuan (Alan) Qi Purdue CS Oct

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

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

Dual variables: Dual Representation for Ridge Regression

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

Generate Kernel Matrix Positive semidefinite Consider Gaussian kernel:

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

Feature Mapping Eigen-problem in feature space

Dual Variables Suppose, we have

Eigen-problem in Feature Space (1)

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

General Case for Non-zero Mean Case Kernel Matrix:

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)

Linear Regression Revisited Let We have

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

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.)

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 :

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

Gaussian Process for Regression Likelihood: Prior: Marginal distribution:

Samples of GP Prior over Functions

Samples of Data Points

Predictive Distribution is a Gaussian distribution with mean and variance:

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

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

Marginal Distribution of Target Values