Homework6-solutions

Homework6-solutions - Neural Networks CAP 6615, Spring 2011...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Neural Networks CAP 6615, Spring 2011 Homework 6 Solutions 1. (Problem 8.15 Haykin) Let u k ij denote the centered counterpart of the ij-th element k ij of the Gram K . Derive the following formula (Schölkopf, 1977): u k ij = k ij m 1 N ∑ m = 1 N Φ T ( x m )Φ( x j )m 1 N ∑ n = 1 N Φ T ( x i )Φ( x n )+ 1 N 2 ∑ m = 1 N ∑ n = 1 N Φ T ( x m )Φ( x n ) . Suggest a compact representation of this relation in matrix form. I'll answer the second question first. Consider this compact representation of the relation: u k ij = ( Φ T ( x i )m u Φ T ) ( Φ( x j )m u Φ ) = ( Φ T ( x i )m 1 N ∑ m = 1 N Φ T ( x m ) )( Φ( x j )m 1 N ∑ n = 1 N Φ( x n ) ) =Φ T ( x i )Φ( x j )m 1 N ∑ m = 1 N Φ T ( x m )Φ( x j )m 1 N Φ T ( x i ) ∑ n = 1 N Φ( x n )m 1 N 2 ∑ m = 1 N ∑ n = 1 N Φ T ( x m )Φ( x n ) = k ij m 1 N ∑ m = 1 N Φ T ( x m )Φ( x j )m 1 N ∑ n = 1 N Φ T ( x i )Φ( x n )+ 1 N 2 ∑ m = 1 N ∑ n = 1 N Φ T ( x m )Φ( x n ) . 2. (Problem 8.16 Haykin) Show that the normalization of eigenvector α of the Gram K is equivalent to the requirement that Eq. (8.109) be satisfied. This problem is a relatively straightforward algebraic task. Equation (8.100) is the key. From this we get 1 = ̃ q T q = ( ∑ i = 1 N α i ϕ( x i ) ) T ( ∑ j = 1 N α i ϕ( x j ) ) = ∑ i = 1 N ∑ j = 1 N ϕ T ( x i )α i α j ϕ( x j )= ∑ i = 1 N ∑ j = 1 N α i α j ϕ T ( x i )ϕ( x j ) = ∑ i = 1 N ∑ j = 1 N α i α j k ( x i , x j )= ∑ i = 1 N ∑ j = 1 N α i α j N ̃ λ[ 8.101 ]=α T α λ . So, since α r T α r λ r = 1 , α r T α r = 1 λ r . 3. (Problem 9.8 Haykin) Using the transformation formula of Eq. (9.40) applied to Eq. (9.37), derive the probability density function of Eq. (9.41). [Note: I believe Haykin's Eq. 9.41 is missing a 1 σ term. Please verify or disprove this.] The X 2 probability density function is defined by...
View Full Document

This note was uploaded on 11/30/2011 for the course CAP 6615 taught by Professor Staff during the Spring '08 term at University of Florida.

Page1 / 5

Homework6-solutions - Neural Networks CAP 6615, Spring 2011...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online