There are actually two equivalent ways to think about PCA The first is

There are actually two equivalent ways to think about

This preview shows page 10 - 13 out of 18 pages.

There are actually two equivalent ways to think about PCA. The first is statistical: we are trying to find a transform that is carefully tuned to the (second-order) statistics of the data. The second perspective, which is what we will adopt in this course, is more geometrical: given a set of vectors, we are trying to find a subspace of a certain dimension that comes closest to containing this set. Specifically, suppose that we have data points x 1 , . . . , x N R D , and want to find the K -dimensional affine space (subspace plus offset) that comes closest to containing them. Here is a picture 4 Example From Chapter 14 of Hastie, Tibshirani, and Friedman 16 4 From Ch. 14 of Tibshirani and Hastie’s Elements of Statistical Learning . 76 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 23:01, November 5, 2019
Image of page 10

Subscribe to view the full document.

Our goal is to find an offset μ R D and a matrix Q with orthonormal columns such that x n μ + n for all n = 1 , . . . , N, for some θ n R K . We cast this as the following optimization prob- lem. Given x 1 , . . . , x N , solve minimize μ , Q , { θ n } N X n =1 k x n - μ - n k 2 2 subject to Q T Q = I . Note that if we fix μ and define e x n = x n - μ , then we can recast the optimization with respect to Q and the θ n as minimize Q , { θ n } N X n =1 k e x n - n k 2 2 subject to Q T Q = I . If f X and Θ denote the matrices whose columns are given by e x 1 , . . . , e x n and θ 1 , . . . , θ n respectively, then we can also write this as minimize Q : D × K Θ : K × N k f X - Q Θ k 2 F subject to Q T Q = I . This is exactly the optimization problem that we looked at previ- ously in our Subspace Approximation Lemma! Thus the solution is given by computing the SVD of f X = U Σ V and then taking as our solution b Q = U K , b Θ = U T K f X , where U K = u 1 u 2 · · · u K contains the first K columns of U . 77 Georgia Tech ECE 6250 Fall 2019; Notes by J. Romberg and M. Davenport. Last updated 23:01, November 5, 2019
Image of page 11
Finally, let us return to the question of how to set μ . For any given μ , the solution for Q and Θ is given by the Subspace Approximation Lemma. This results in setting θ n = Q T ( x n - μ ) . Plugging this in for θ n in our objective function, we have that x n - μ - n = x n - μ - QQ T ( x n - μ ) = ( I - QQ T )( x n - μ ) . Hence, the problem of selecting μ reduces to the optimization prob- lem minimize μ N X n =1 k ( I - QQ T )( x n - μ ) k 2 2 The vector μ is unconstrained; we can solve for the optimal μ by taking a gradient and setting it equal to zero. To make this easier, note that k ( I - QQ T )( x n - μ ) k 2 2 = ( x n - μ ) T ( I - QQ T )( x n - μ ) by simply expanding out the norm squared as an inner product and then using the fact that I - QQ T is a projector, i.e., it is symmetric and ( I - QQ T ) 2 = I - QQ T . Thus, by taking a gradient and setting it equal to zero we have 0 = - 2 N X n =1 ( I - QQ T )( x n - μ ) = - 2( I - QQ T ) N X n =1 x n !
Image of page 12

Subscribe to view the full document.

Image of page 13
  • Fall '08
  • Staff

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

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes