Ref.
Exercises
[1]
2.2, 2.12 (d)(e)(g), 2.18, 2.24, 3.3, 3.18, 3.19(a)
Matlab Assignment
Problem 1. Let the set S be described by,
.
(a) Use Matlab to draw the set S and investigate its convexity.
(b) Show your conclusion in part (a) theoretically.
Problem 2. Let Q1 and Q2 be arbitrary n × n symmetric matrices (n > 2).
(a) Use Matlab to draw the set and investigate its convexity.
(b) Extra point: Can you show your conclusion in part (a) theoretically?
Problem 3. Let A be a real m×n matrix with a singular value decomposition given by A = UΣV T (as discussed in class). For a positive integer k ≤ min{m,n}, we let Ak denote an m × n matrix which is an
“approximation” of the matrix A obtained from its top k singular values and singular vectors, i.e.,
Ak = UkΣkVkT,
where Uk has the first k columns of U, Vk has the first k columns of V , and Σk is the upper left k × k block of Σ.
(a) To provide a good approximation for A, consider the cost function kA − Xk2 where X is restricted to be an m × n matrix with rank(X) ≤ k. It can be shown that Ak is the minimizer of the cost function kA − Xk2. Download the file HajiFirouz.jpg. Read this file in Matlab by typing:
A=imread(’HajiFirouz.jpg’); A=im2double(A) ;
A=rgb2gray(A) ;
The result is a 395 × 665 matrix A, with each entry representing a single pixel in the picture with a number between 0 and 1.
For different values of k, use Matlab to compute Ak, construct a compressed image with Ak (You can used the command imwrite), and report the value of kA − Akk2.
(b) Based on your experiments in part (a), provide a good compressed image for HajiFirouz and explain your interpretations.
