KpNormDual - Revision history

Nathaniel: typo

2016-05-25T14:45:46Z

typo

← Older revision		Revision as of 14:45, 25 May 2016
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	<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the [[kpNorm\|(k,p)-norm]] of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is		<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the [[kpNorm\|(k,p)-norm]] of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is
	: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>		: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>
	where $(x_1^\downarrow,x_2^\downarrow,\ldots,x_n^\downarrow)$ is a rearrangement of the vector $x$ with the property that $\|x_1^\downarrow\| \geq \|x_2^\downarrow\| \geq \cdots \geq \|x_n^\downarrow\|$. Similarly, the (k,p)-norm of a matrix is the (k,p)-norm of its vector of singular values. ~~Thus~~ function computes the dual of this norm, which is fairly complicated and was derived in<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. This function works with both full and sparse vectors and matrices.		where $(x_1^\downarrow,x_2^\downarrow,\ldots,x_n^\downarrow)$ is a rearrangement of the vector $x$ with the property that $\|x_1^\downarrow\| \geq \|x_2^\downarrow\| \geq \cdots \geq \|x_n^\downarrow\|$. Similarly, the (k,p)-norm of a matrix is the (k,p)-norm of its vector of singular values. This function computes the dual of this norm, which is fairly complicated and was derived in<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. This function works with both full and sparse vectors and matrices.

	==Syntax==		==Syntax==

Nathaniel: cvx-safe

2015-06-24T18:57:03Z

cvx-safe

← Older revision		Revision as of 18:57, 24 June 2015
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	\|rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=~~April 3~~, ~~2013~~		\|upd=June 24, 2015
	\|cvx=no}}		\|cvx=yes (convex)}}
	<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the [[kpNorm\|(k,p)-norm]] of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is		<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the [[kpNorm\|(k,p)-norm]] of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is
	: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>		: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>

Nathaniel at 16:46, 24 December 2014

2014-12-24T16:46:05Z

← Older revision		Revision as of 16:46, 24 December 2014
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	\|upd=April 3, 2013		\|upd=April 3, 2013
	\|cvx=no}}		\|cvx=no}}
	<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the (k,p)-norm of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is		<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the [[kpNorm\|(k,p)-norm]] of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is
	: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>		: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>
	where $(x_1^\downarrow,x_2^\downarrow,\ldots,x_n^\downarrow)$ is a rearrangement of the vector $x$ with the property that $\|x_1^\downarrow\| \geq \|x_2^\downarrow\| \geq \cdots \geq \|x_n^\downarrow\|$. Thus function computes the dual of this norm, which is fairly complicated and was derived in<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. This function works with both full and sparse vectors and matrices.		where $(x_1^\downarrow,x_2^\downarrow,\ldots,x_n^\downarrow)$ is a rearrangement of the vector $x$ with the property that $\|x_1^\downarrow\| \geq \|x_2^\downarrow\| \geq \cdots \geq \|x_n^\downarrow\|$. Similarly, the (k,p)-norm of a matrix is the (k,p)-norm of its vector of singular values. Thus function computes the dual of this norm, which is fairly complicated and was derived in<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. This function works with both full and sparse vectors and matrices.

	==Syntax==		==Syntax==

Nathaniel: added CVX parameter

2014-12-24T16:45:04Z

added CVX parameter

← Older revision		Revision as of 16:45, 24 December 2014
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	{{Function		{{Function
	\|name=kpNormDual		\|name=kpNormDual
	\|desc=Computes the [[dual of the (k,p)-norm]] of a vector or matrix		\|desc=Computes the dual of the (k,p)-norm of a vector or matrix
	\|rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=April 3, 2013		\|upd=April 3, 2013
	\|v=~~0.50~~}}		\|cvx=no}}
	<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the [[dual of the (k,p)-norm]]<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref> ~~of a vector or matrix~~. It works with both full and sparse vectors and matrices.		<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the dual of the (k,p)-norm of a vector or matrix. More explicitly, the (k,p)-norm of a vector $x = (x_1,x_2,\ldots,x_n)$ is
			: <math>\\|x\\|_{(k,p)} := \left(\sum_{j=1}^k \big\|x_i^\downarrow\big\|^p \right)^{1/p},</math>
			where $(x_1^\downarrow,x_2^\downarrow,\ldots,x_n^\downarrow)$ is a rearrangement of the vector $x$ with the property that $\|x_1^\downarrow\| \geq \|x_2^\downarrow\| \geq \cdots \geq \|x_n^\downarrow\|$. Thus function computes the dual of this norm, which is fairly complicated and was derived in<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. This function works with both full and sparse vectors and matrices.

	==Syntax==		==Syntax==

Nathaniel at 16:35, 22 September 2014

2014-09-22T16:35:28Z

← Older revision		Revision as of 16:35, 22 September 2014
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	\|name=kpNormDual		\|name=kpNormDual
	\|desc=Computes the [[dual of the (k,p)-norm]] of a vector or matrix		\|desc=Computes the [[dual of the (k,p)-norm]] of a vector or matrix
	\|~~req~~=[[kpNorm]]		\|rel=[[kpNorm]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	~~\|rel=~~[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=April 3, 2013		\|upd=April 3, 2013
	\|v=1.02}}		\|v=0.50}}
	<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the [[dual of the (k,p)-norm]]<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref> of a vector or matrix. It works with both full and sparse vectors and matrices.		<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the [[dual of the (k,p)-norm]]<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref> of a vector or matrix. It works with both full and sparse vectors and matrices.

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	===A simple 4-by-4 example===		===A simple 4-by-4 example===
	The (k,p)-norm of a matrix when k = 1 is simply the operator norm. The dual of the operator norm is the trace norm, so when k = 1 this function just returns the trace norm (regardless of p):		The (k,p)-norm of a matrix when k = 1 is simply the operator norm. The dual of the operator norm is the trace norm, so when k = 1 this function just returns the trace norm (regardless of p):
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> X = [1 1 1 1;1 2 3 4;1 4 9 16;1 8 27 64];		>> X = [1 1 1 1;1 2 3 4;1 4 9 16;1 8 27 64];
	>> [kpNormDual(X,1,1), ~~[[TraceNorm\|~~TraceNorm(X)]]]		>> [kpNormDual(X,1,1), TraceNorm(X)]

	ans =		ans =

	77.0015 77.0015		77.0015 77.0015
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	Similarly, if <tt>K = min(size(X))</tt> and <tt>P = 2</tt> then <tt>kpNorm(X,K,P)</tt> is the Frobenius norm, which is its own dual. Thus <tt>kpNormDual(X,K,2)</tt> decreases from the trace norm of <tt>X</tt> to its Frobenius norm as <tt>K</tt> increases:		Similarly, if <tt>K = min(size(X))</tt> and <tt>P = 2</tt> then <tt>kpNorm(X,K,P)</tt> is the Frobenius norm, which is its own dual. Thus <tt>kpNormDual(X,K,2)</tt> decreases from the trace norm of <tt>X</tt> to its Frobenius norm as <tt>K</tt> increases:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> [kpNormDual(X,1,2), TraceNorm(X)]		>> [kpNormDual(X,1,2), TraceNorm(X)]

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	72.6498 72.6498		72.6498 72.6498
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

			{{SourceCode\|name=kpNormDual}}

	==References==		==References==
	<references />		<references />

Nathaniel: Updated to v1.02

2013-06-25T21:42:33Z

Updated to v1.02

← Older revision		Revision as of 21:42, 25 June 2013
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	{{Function		{{Function
	\|name=kpNormDual		\|name=kpNormDual
	\|desc=Computes the [[dual of the (k,p)-norm]] of ~~an operator~~		\|desc=Computes the [[dual of the (k,p)-norm]] of a vector or matrix
	\|req=[[kpNorm]]		\|req=[[kpNorm]]
	\|rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|upd=~~December~~ 3, ~~2012~~		\|upd=April 3, 2013
	\|v=1.00}}		\|v=1.02}}
	<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the [[dual of the (k,p)-norm]]<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. It works with both full and sparse matrices.		<tt>'''kpNormDual'''</tt> is a [[List of functions\|function]] that computes the [[dual of the (k,p)-norm]]<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref> of a vector or matrix. It works with both full and sparse vectors and matrices.

	==Syntax==		==Syntax==
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	==Argument descriptions==		==Argument descriptions==
	* <tt>X</tt>: ~~An operator~~ to have its norm computed.		* <tt>X</tt>: A vector or matrix to have its norm computed.
	* <tt>K</tt>: A positive integer.		* <tt>K</tt>: A positive integer.
	* <tt>P</tt>: A real number ≥ 1, or <tt>Inf</tt>.		* <tt>P</tt>: A real number ≥ 1, or <tt>Inf</tt>.
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	==Examples==		==Examples==
	===A simple 4-by-4 example===		===A simple 4-by-4 example===
	The (k,p)-norm when k = 1 is simply the operator norm. The dual of the operator norm is the trace norm, so when k = 1 this function just returns the trace norm (regardless of p):		The (k,p)-norm of a matrix when k = 1 is simply the operator norm. The dual of the operator norm is the trace norm, so when k = 1 this function just returns the trace norm (regardless of p):
	<pre<noinclude></noinclude>>		<pre<noinclude></noinclude>>
	>> X = [1 1 1 1;1 2 3 4;1 4 9 16;1 8 27 64];		>> X = [1 1 1 1;1 2 3 4;1 4 9 16;1 8 27 64];

Nathaniel: Created page with "{{Function |name=kpNormDual |desc=Computes the dual of the (k,p)-norm of an operator |req=kpNorm |rel=KyFanNorm
SchattenNorm
TraceNorm |upd=Decem..."

2012-12-03T15:45:14Z

Created page with "{{Function |name=kpNormDual |desc=Computes the dual of the (k,p)-norm of an operator |req=kpNorm |rel=KyFanNorm<br />SchattenNorm<br />TraceNorm |upd=Decem..."

New page

{{Function
|name=kpNormDual
|desc=Computes the [[dual of the (k,p)-norm]] of an operator
|req=[[kpNorm]]
|rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
|upd=December 3, 2012
|v=1.00}}
<tt>'''kpNormDual'''</tt> is a [[List of functions|function]] that computes the [[dual of the (k,p)-norm]]<ref>G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. ''Proc. Amer. Math. Soc.'', 95:331–337, 1985.</ref>. It works with both full and sparse matrices.

==Syntax==
* <tt>NRM = kpNormDual(X,K,P)</tt>

==Argument descriptions==
* <tt>X</tt>: An operator to have its norm computed.
* <tt>K</tt>: A positive integer.
* <tt>P</tt>: A real number ≥ 1, or <tt>Inf</tt>.

==Examples==
===A simple 4-by-4 example===
The (k,p)-norm when k = 1 is simply the operator norm. The dual of the operator norm is the trace norm, so when k = 1 this function just returns the trace norm (regardless of p):
<pre<noinclude></noinclude>>
>> X = [1 1 1 1;1 2 3 4;1 4 9 16;1 8 27 64];
>> [kpNormDual(X,1,1), [[TraceNorm|TraceNorm(X)]]]

ans =

77.0015 77.0015
</pre<noinclude></noinclude>>

Similarly, if <tt>K = min(size(X))</tt> and <tt>P = 2</tt> then <tt>kpNorm(X,K,P)</tt> is the Frobenius norm, which is its own dual. Thus <tt>kpNormDual(X,K,2)</tt> decreases from the trace norm of <tt>X</tt> to its Frobenius norm as <tt>K</tt> increases:
<pre<noinclude></noinclude>>
>> [kpNormDual(X,1,2), TraceNorm(X)]

ans =

77.0015 77.0015

>> kpNormDual(X,2,2)

ans =

72.6903

>> kpNormDual(X,3,2)

ans =

72.6505

>> [kpNormDual(X,4,2), norm(X,'fro')]

ans =

72.6498 72.6498
</pre<noinclude></noinclude>>

==References==
<references />