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I recently came across this notation, without explanation, in a paper:


From the context, I know that $\mathbf{W}$ is a matrix, which could be any size, and that $||\mathbf{W}||_{2,1}$ must be a scalar. Can anyone tell me what this means, or just what it's called, so I can look it up?

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Guess 1: norm as a transformation from $l^1$ to $l^2$. Guess 2: norm as an operator on Sobolev space $W_{1,2}$. Perhaps the rest of the paper suggests the area more specifically than this... –  Gerald Edgar Jun 9 '13 at 16:46
Perhaps it means the maximum value of the $L_1$-norm of $Wx$ divided by the $L_2$-norm of $x$ over all nonzero $x$. –  Will Sawin Jun 9 '13 at 16:47
Will Sawin, I see Wikipedia uses this notation with that meaning (en.wikipedia.org/wiki/Matrix_norm, a bit past the halfway point in the "Induced norm" section). I don't know how I failed to find this before, but I guess this is probably it. –  Tom Future Jun 9 '13 at 16:56
Similar questions about "mixed-norms" have been previously answered either on MO or M.SE; voting to close this one. It can mean many things; one common and easy to remember instance of $\|X\|_{p,q}$ is that it is the $p$-norm of the vector formed by computing $q$-norms of the columns of $X$.... –  Suvrit Jun 9 '13 at 18:16

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