Solution 10 Computational Intelligence Lab 2012

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Problem 1

1
a) rank=4, eigenvalues = -1,1,2,2 => spectral norm = 2
b) rank=2, eigenvalues = 0,3,3 => spectral norm = 3
2
rank(L) = 2
card(S) = 3
F(L,S) = 10.9994 + 3 = 13.9994 (no idea how to compute nuclear norm of L by hand...)

Problem 2

1
  • 1-norm sums up the absolute value of all elements -> small if only few elements are nonzero
  • large matrix with one very large value
  • large matrix filled with small values
2
  • nuclear norm = sum of singular values -> small if only few singular values are nonzero -> small rank
  • rank 1 matrix with a very high singular value
  • full rank matrix with all very small singular values (nearly singular matrix)
3
  • in general, sparse means number of elements are of order O(sqrt(m*n))