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Thread: Arg Max notation?

  1. #1 Arg Max notation? 
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    So I'm having trouble understanding the max notation and exactly what it means.


    I have a problem I'm working and this is what it is saying



    What is the interpretation of this in english, I understand this much.

    L1 norm of A = to the max of ||Ax|| when ||x|| = 1 , but what is a good way of thinking about it.
    d
    I've also seen the notation used with arg max, I assume they are the same. I'm just having a hard time understanding the interpretation of this.


    Last edited by GenerationE; September 20th, 2011 at 08:21 PM.
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  3. #2  
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    The notation is used for any norm, not just L1. I presume you know what A and x mean in this context. ||x|| is the norm of a vector x. Consider the subset consisting of all vectors in the space with ||x||=1, and then all the images (Ax) of these vectors under the operator A. max||Ax|| of these is the norm of A.

    The third term (max j) looks screwed up.


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  4. #3  
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    A is a matrix or transformation and x is a vector. Their image is another vector.

    So the l1 norm of a vector would then be column that gives the maximum l1 norm.

    For instance:

    A = (1 2; 3 4)

    L1 norm of A = 2 + 4 = 6 ?

    Is that a correct assumption? To say that the norm would be sum of the column with the largest values? since the norm of x is 1 we are selecting the largest column.
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    I haven't examined it in detail, but it looks OK so far. Have you looked at cases where the matrix elements are some positive and some negative?

    I think you need to look at the max j term - it seems to have typos.
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  6. #5  
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    Quote Originally Posted by GenerationE View Post
    I think I fixed it, the j term wasn't supposed to be multiplied. I have it right on paper, I just entered in incorrectly on here.

    I expanded this to the infinity norm of A and I got that the the Infinity norm of a matrix is the sum of the row with the largest values. It's interesting that these appear to be inverses of each other. I'm now working on the L2 norm or a matrix but it's much more difficult than the other two.
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  7. #6  
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    It still doesn't look right, the sum is over j and you have max over j outside the sum. Should it be max over i?
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