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

Info icon This 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 Document Right 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

  • Spring '08
  • Staff
  • Probability theory, probability density function, Cumulative distribution function, Haykin, Neural Networks CAP

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern