stat441w18/summary 1

Group members

Faith Lee

Jacov Lisulov

Shiwei Gong

Introduction and problem motivation

In classification problems, kernel methods are used for pattern analysis and require only user-specified kernel. Kernel methods can be thought of as instance-based methods where they learn the i-th training examples are "remembered" to learn for corresponding weights. Prediction on untrained examples are then treated with a similarity function, k (also called a kernel between this untrained example and each of the training inputs. In short, a similarity function measures similarity between two objects. By conventional notation, we have that

[math]\displaystyle{ f(x; \alpha) = \sum_{i = 1}^{N} \alpha_i k(x, x_i) }[/math] where k is a kernel function, [math]\displaystyle{ k(x, x') \approx \sum_{j = i}^{D} Z(X; W_j)Z(X'; W_j) }[/math]

and [math]\displaystyle{ \alpha }[/math] are the corresponding weights.

An example of a kernel method is the support vector machine. Kernel methods provides a means of approximating a non-linear function or decision boundary. However, the problem of using kernel methods are that:

1) It scales poorly with the size of the dataset

2) Requires computation of matrices (and inverses)

3) Solving linear system of equations

In this paper, the authors propose