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Looking for tighter bound on symmetric PSD matrices products

  1. Sep 20, 2015 #1
    1. The problem statement, all variables and given/known data
    Let K and L be symmetric PSD matrices of size N*N, with all entries in [0,1]. Let i be any number in 1...N and K’, L’ be two new symmetric PSD matrices, each with only row i and column i different from K and L. I would like to obtain an upper bound of the equation below: where .∗ is an element-wise multiply.


    2. Relevant equations
    |[sum(K' .∗ L') − sum(K .∗ L)] − (2/N)*[sum(K'L'') − sum(KL)] + (1/N*N)*[sum(K')sum(L') − sum(K)sum(L)]|

    See attached for equation in LaTex/PDF.

    3. The attempt at a solution
    Using simple triangular inequality and bounding the three square brackets respectively,
    I can bound the above with 12N − 13.
    However, this is extremely loose.
    Empirical experiments show that the constant coefficient should be much lower, probably closer to 1N.

    Can you suggest any linear algebra properties connecting the equation's parts - which may help me get a tighter bound?
     

    Attached Files:

  2. jcsd
  3. Sep 25, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
  4. Sep 25, 2015 #3

    Ray Vickson

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    Homework Helper

    Does ##\sum(A)## mean ##\sum_i \sum_j a_{ij}## for matrix ##A = (a_{ij})##? When you say that each of ##K', L'##have only row ##i## and column ##i## different from ##K,L##, do you mean that ##K'## has both its row and column ##i## different from ##K## and that ##L'## has both its row and its column ##i## different from ##L##? I think that is what you mean, but it is worth checking. Are all the elements of ##K', L'## also in ##[0,1]##?
     
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