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Least Square Estimator

  1. Mar 31, 2015 #1

    bhobba

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    I recently came across the following interesting problem.

    Suppose A = BC where A,B, and C are matrices. We know a ton of A's and their corresponding C's. We want the least square estimator of B.

    When A and C are vectors the solution is well known.

    But what is the solution when they are matrices?

    Thanks
    Bill
     
  2. jcsd
  3. Mar 31, 2015 #2

    Stephen Tashi

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    How do you intend to define the "error" between the observed and predicted values? Until that is defined, "least squares" doesn't describe a specific criteria.
     
  4. Mar 31, 2015 #3

    bhobba

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    The matrix L2 norm ie the B that minimises ∑ ||Ai - BCi||^2 where Ai and Ci are the known A C matrix pairs. By matrix L2 norm I mean the generalisation of the usual vector norm ie the square root of the sum of the squares of the matrix elements.

    It grew out of the following paper:
    http://www.cv-foundation.org/openac...t_Direct_Super-Resolution_2013_ICCV_paper.pdf

    See equation 2.

    Thanks
    Bill
     
  5. Mar 31, 2015 #4

    WWGD

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    Isn't there an equivalent of a perp projection operator in your space of matrices ?If this space is a Hilbert space, then, AFAIK, the general solution to this problem in a Hilbert space is the ortho. projection of B onto the subspace spanned by A,C.
     
  6. Mar 31, 2015 #5

    bhobba

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    Yes there is - its the trace. I will think about that one.

    Thanks
    Bill
     
  7. Apr 1, 2015 #6

    bhobba

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    Hi Guys

    Thanks for all the help.

    Finally nutted it out. As usual I was on the wrong track. It's simply a matter by blocking the problem and reducing it to a number of ordinary least squares problems. Break B into rows Bj so you get the usual least squares problems ||Aji - BjCi||^2. The minimum is the minimum of each of these separate problems.

    Thanks
    Bill
     
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