Comments on: Not up to norm http://bit-player.org/2009/not-up-to-norm An amateur's outlook on computation and mathematics. Mon, 15 Mar 2010 06:21:10 +0000 http://wordpress.org/?v=2.6.3 By: David S. Mazel http://bit-player.org/2009/not-up-to-norm#comment-2162 David S. Mazel Wed, 24 Jun 2009 02:34:47 +0000 http://bit-player.org/?p=384#comment-2162 Kudos to you, Brian, for writing about your experiences in trying to express mathematical concepts as clearly as possible. It takes a certain amount of humility and comfort with oneself to express the thoughts you have shown here. Bravo! That said, I have to quote, as best I can, Albert Einstein: "Make everything as simple as possible but not simpler." I think you may have simplified just a little too much. :-) Kudos to you, Brian, for writing about your experiences in trying to express mathematical concepts as clearly as possible. It takes a certain amount of humility and comfort with oneself to express the thoughts you have shown here. Bravo!

That said, I have to quote, as best I can, Albert Einstein:

“Make everything as simple as possible but not simpler.”

I think you may have simplified just a little too much. :-)

]]> By: Jim Ward http://bit-player.org/2009/not-up-to-norm#comment-2160 Jim Ward Tue, 23 Jun 2009 14:10:56 +0000 http://bit-player.org/?p=384#comment-2160 If you type y = x^x into Wolfram alpha, the real part has a nice continuous graph at 0. If you type y = x^x into Wolfram alpha, the real part has a nice continuous graph at 0.

]]> By: JK http://bit-player.org/2009/not-up-to-norm#comment-2157 JK Sun, 21 Jun 2009 12:18:01 +0000 http://bit-player.org/?p=384#comment-2157 The L_p norm can be defined for any positive real p > 0. Doesn't defining L_0(v) = lim_{p -> 0}(L_p(v)) give what we want? The L_p norm can be defined for any positive real p > 0.

Doesn’t defining L_0(v) = lim_{p -> 0}(L_p(v)) give what we want?

]]> By: Igor Carron http://bit-player.org/2009/not-up-to-norm#comment-2156 Igor Carron Sun, 21 Jun 2009 07:12:29 +0000 http://bit-player.org/?p=384#comment-2156 Brian, You really want to deal with underdetermined systems (i.e systems where you have more unknowns than equations). In this case, you have an infinity of solutions. The point of the solvers used in compressive sensing is to find the sparsest solution of all these solutions. In the L2 sense, you are looking with the one with the smallest norm, here you are looking for the one with the smaller number of coefficients. With regards to L_0, it really is just a semi-norm, an extension of what is seen for p >=1. The definition of l_0 is specifically that the cardinal of the set (i.e. number of elements not equal to zero). By the way, L_1 is really the sum of the absolute values of the elements in the vector. Since trying to compute the problem using l_0 is NP-Hard, i.e. takes forever, the neat trick was to find that replacing l_0 by l_1 worked. Hope it helps, Cheers, Igor. Brian,

You really want to deal with underdetermined systems (i.e systems where you have more unknowns than equations). In this case, you have an infinity of solutions. The point of the solvers used in compressive sensing is to find the sparsest solution of all these solutions. In the L2 sense, you are looking with the one with the smallest norm, here you are looking for the one with the smaller number of coefficients.

With regards to L_0, it really is just a semi-norm, an extension of what is seen for p >=1. The definition of l_0 is specifically that the cardinal of the set (i.e. number of elements not equal to zero). By the way, L_1 is really the sum of the absolute values of the elements in the vector.

Since trying to compute the problem using l_0 is NP-Hard, i.e. takes forever, the neat trick was to find that replacing l_0 by l_1 worked.

Hope it helps,

Cheers,

Igor.

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