KpNorm - Revision history

Nathaniel: cvx-safe

2015-06-24T18:56:37Z

cvx-safe

← Older revision		Revision as of 18:56, 24 June 2015
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	\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=~~October 22~~, ~~2014~~		\|upd=June 24, 2015
	\|cvx=no}}		\|cvx=yes (convex)}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the (k,p)-norm of a vector or matrix, defined as follows:		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the (k,p)-norm of a vector or matrix, defined as follows:
	: <math>\\|v\\|_{k,p} := \left(\sum_{j=1}^k \|v_j^{\downarrow}\|^p \right)^{1/p}, \quad \\|X\\|_{k,p} := \left( \sum_{j=1}^k \sigma_j^{\downarrow}(X)^p \right)^{1/p}.</math>		: <math>\\|v\\|_{k,p} := \left(\sum_{j=1}^k \|v_j^{\downarrow}\|^p \right)^{1/p}, \quad \\|X\\|_{k,p} := \left( \sum_{j=1}^k \sigma_j^{\downarrow}(X)^p \right)^{1/p}.</math>

Nathaniel: added CVX parameter

2014-12-24T16:47:18Z

added CVX parameter

← Older revision		Revision as of 16:47, 24 December 2014
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	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=October 22, 2014		\|upd=October 22, 2014
	\|v=~~0.50~~}}		\|cvx=no}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the (k,p)-norm of a vector or matrix, defined as follows:		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the (k,p)-norm of a vector or matrix, defined as follows:
	: <math>\\|v\\|_{k,p} := \left(\sum_{j=1}^k \|v_j^{\downarrow}\|^p \right)^{1/p}, \quad \\|X\\|_{k,p} := \left( \sum_{j=1}^k \sigma_j^{\downarrow}(X)^p \right)^{1/p}.</math>		: <math>\\|v\\|_{k,p} := \left(\sum_{j=1}^k \|v_j^{\downarrow}\|^p \right)^{1/p}, \quad \\|X\\|_{k,p} := \left( \sum_{j=1}^k \sigma_j^{\downarrow}(X)^p \right)^{1/p}.</math>
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	==Examples==		==Examples==
	===Generalizes the operator, trace, Ky Fan, and Schatten norms===		===Generalizes the operator, trace, Ky Fan, and Schatten norms===
	The (<tt>K</tt>,<tt>P</tt>)-norm of a matrix is simply the usual [[operator norm]] when <tt>K = 1</tt> or <tt>P = Inf</tt>:		The (<tt>K</tt>,<tt>P</tt>)-norm of a matrix is simply the usual operator norm when <tt>K = 1</tt> or <tt>P = Inf</tt>:
	<syntaxhighlight>		<syntaxhighlight>
	>> X = rand(3);		>> X = rand(3);
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	</syntaxhighlight>		</syntaxhighlight>

	When <tt>P = 1</tt> and <tt>K</tt> is the size of <tt>X</tt>, this norm reduces to the [[trace norm]]:		When <tt>P = 1</tt> and <tt>K</tt> is the size of <tt>X</tt>, this norm reduces to the trace norm:
	<syntaxhighlight>		<syntaxhighlight>
	>> [kpNorm(X,3,1), TraceNorm(X)]		>> [kpNorm(X,3,1), TraceNorm(X)]
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	</syntaxhighlight>		</syntaxhighlight>

	More generally, when <tt>P = 1</tt> this norm reduces to the ~~[[Ky Fan norm\|~~Ky Fan <tt>K</tt>-norm]]:		More generally, when <tt>P = 1</tt> this norm reduces to the Ky Fan <tt>K</tt>-norm:
	<syntaxhighlight>		<syntaxhighlight>
	>> [kpNorm(X,2,1), KyFanNorm(X,2)]		>> [kpNorm(X,2,1), KyFanNorm(X,2)]
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	</syntaxhighlight>		</syntaxhighlight>

	Similarly, when <tt>K = min(size(X))</tt> this norm reduces to the ~~[[Schatten norm\|~~Schatten <tt>P</tt>-norm]]:		Similarly, when <tt>K = min(size(X))</tt> this norm reduces to the Schatten <tt>P</tt>-norm:
	<syntaxhighlight>		<syntaxhighlight>
	>> [kpNorm(X,3,4), SchattenNorm(X,4)]		>> [kpNorm(X,3,4), SchattenNorm(X,4)]

Nathaniel: Added definition

2014-10-23T13:54:34Z

Added definition

← Older revision		Revision as of 13:54, 23 October 2014
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	\|upd=October 22, 2014		\|upd=October 22, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of a vector or matrix. In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. It works with both full and sparse vectors and matrices.		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the (k,p)-norm of a vector or matrix, defined as follows:
			: <math>\\|v\\|_{k,p} := \left(\sum_{j=1}^k \|v_j^{\downarrow}\|^p \right)^{1/p}, \quad \\|X\\|_{k,p} := \left( \sum_{j=1}^k \sigma_j^{\downarrow}(X)^p \right)^{1/p}.</math>
			In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. This norm generalizes many well-known norms including the vector p-norms, the operator norm (when k = 1), the trace norm (when p = 1 and k is the size of X), the Schatten p-norms (when k is the size of X), and the Ky Fan k-norms (when p = 1). It works with both full and sparse vectors and matrices.

	==Syntax==		==Syntax==

Nathaniel: Added CVX functionality

2014-10-23T02:07:32Z

Added CVX functionality

← Older revision		Revision as of 02:07, 23 October 2014
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	{{Function		{{Function
	\|name=kpNorm		\|name=kpNorm
	\|desc=Computes the [[(k,p)-norm]] of a vector or matrix		\|desc=Computes the (k,p)-norm of a vector or matrix
	\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|cat=[[List of functions#Norms\|Norms]]		\|cat=[[List of functions#Norms\|Norms]]
	\|upd=~~March 9~~, ~~2013~~		\|upd=October 22, 2014
	\|v=0.50}}		\|v=0.50}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of a vector or matrix. In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. It works with both full and sparse vectors and matrices.		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of a vector or matrix. In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. It works with both full and sparse vectors and matrices.

Nathaniel at 16:33, 22 September 2014

2014-09-22T16:33:27Z

← Older revision		Revision as of 16:33, 22 September 2014
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	\|desc=Computes the [[(k,p)-norm]] of a vector or matrix		\|desc=Computes the [[(k,p)-norm]] of a vector or matrix
	\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
			\|cat=[[List of functions#Norms\|Norms]]
	\|upd=March 9, 2013		\|upd=March 9, 2013
	\|v=1.02}}		\|v=0.50}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of a vector or matrix. In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. It works with both full and sparse vectors and matrices.		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of a vector or matrix. In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. It works with both full and sparse vectors and matrices.

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	===Generalizes the operator, trace, Ky Fan, and Schatten norms===		===Generalizes the operator, trace, Ky Fan, and Schatten norms===
	The (<tt>K</tt>,<tt>P</tt>)-norm of a matrix is simply the usual [[operator norm]] when <tt>K = 1</tt> or <tt>P = Inf</tt>:		The (<tt>K</tt>,<tt>P</tt>)-norm of a matrix is simply the usual [[operator norm]] when <tt>K = 1</tt> or <tt>P = Inf</tt>:
	<~~pre~~>		<syntaxhighlight>
	>> X = rand(3);		>> X = rand(3);
	>> [norm(X), kpNorm(X,1,Inf), kpNorm(X,2,Inf), kpNorm(X,3,Inf), kpNorm(X,1,5)]		>> [norm(X), kpNorm(X,1,Inf), kpNorm(X,2,Inf), kpNorm(X,3,Inf), kpNorm(X,1,5)]
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	1.0673 1.0673 1.0673 1.0673 1.0673		1.0673 1.0673 1.0673 1.0673 1.0673
	</~~pre~~>		</syntaxhighlight>

	When <tt>P = 1</tt> and <tt>K</tt> is the size of <tt>X</tt>, this norm reduces to the [[trace norm]]:		When <tt>P = 1</tt> and <tt>K</tt> is the size of <tt>X</tt>, this norm reduces to the [[trace norm]]:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> [kpNorm(X,3,1), ~~[[TraceNorm\|~~TraceNorm(X)]]]		>> [kpNorm(X,3,1), TraceNorm(X)]

	ans =		ans =

	1.6482 1.6482		1.6482 1.6482
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	More generally, when <tt>P = 1</tt> this norm reduces to the [[Ky Fan norm\|Ky Fan <tt>K</tt>-norm]]:		More generally, when <tt>P = 1</tt> this norm reduces to the [[Ky Fan norm\|Ky Fan <tt>K</tt>-norm]]:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> [kpNorm(X,2,1), ~~[[KyFanNorm\|~~KyFanNorm(X,2)]]]		>> [kpNorm(X,2,1), KyFanNorm(X,2)]

	ans =		ans =

	1.5816 1.5816		1.5816 1.5816
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

	Similarly, when <tt>K = min(size(X))</tt> this norm reduces to the [[Schatten norm\|Schatten <tt>P</tt>-norm]]:		Similarly, when <tt>K = min(size(X))</tt> this norm reduces to the [[Schatten norm\|Schatten <tt>P</tt>-norm]]:
	<~~pre<noinclude></noinclude>~~>		<syntaxhighlight>
	>> [kpNorm(X,3,4), ~~[[SchattenNorm\|~~SchattenNorm(X,4)]]]		>> [kpNorm(X,3,4), SchattenNorm(X,4)]

	ans =		ans =

	1.0814 1.0814		1.0814 1.0814
	</~~pre<noinclude></noinclude>~~>		</syntaxhighlight>

			{{SourceCode\|name=kpNorm}}

Nathaniel: Updated to v1.02

2013-06-25T21:40:27Z

Updated to v1.02

← Older revision		Revision as of 21:40, 25 June 2013
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	{{Function		{{Function
	\|name=kpNorm		\|name=kpNorm
	\|desc=Computes the [[(k,p)-norm]] of ~~an operator~~		\|desc=Computes the [[(k,p)-norm]] of a vector or matrix
	\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|upd=~~December 3~~, ~~2012~~		\|upd=March 9, 2013
	\|v=1.01}}		\|v=1.02}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of ~~an operator~~ (~~i.e~~., the p-norm of the vector of its k largest singular values). It works with both full and sparse matrices.		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of a vector or matrix. In the case of a vector, this is the p-norm of the vector's k largest (in magnitude) entries. In the case of a matrix, this is the p-norm of the vector of its k largest singular values. 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 (<tt>K</tt>,<tt>P</tt>)-norm computed.		* <tt>X</tt>: A vector or matrix to have its (<tt>K</tt>,<tt>P</tt>)-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==
	===Generalizes the operator, trace, Ky Fan, and Schatten norms===		===Generalizes the operator, trace, Ky Fan, and Schatten norms===
	The (<tt>K</tt>,<tt>P</tt>)-norm is simply the usual [[operator norm]] when <tt>K = 1</tt> or <tt>P = Inf</tt>:		The (<tt>K</tt>,<tt>P</tt>)-norm of a matrix is simply the usual [[operator norm]] when <tt>K = 1</tt> or <tt>P = Inf</tt>:
	<pre>		<pre>
	>> X = rand(3);		>> X = rand(3);

Nathaniel at 15:31, 3 December 2012

2012-12-03T15:31:39Z

← Older revision		Revision as of 15:31, 3 December 2012
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	\|name=kpNorm		\|name=kpNorm
	\|desc=Computes the [[(k,p)-norm]] of an operator		\|desc=Computes the [[(k,p)-norm]] of an operator
	\|rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[kpNormDual]]<br />[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|upd=December 3, 2012		\|upd=December 3, 2012
	\|v=1.01}}		\|v=1.01}}

Nathaniel: Uploaded v1.01

2012-12-03T15:12:20Z

Uploaded v1.01

← Older revision		Revision as of 15:12, 3 December 2012
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	\|desc=Computes the [[(k,p)-norm]] of an operator		\|desc=Computes the [[(k,p)-norm]] of an operator
	\|rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]		\|rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]]
	\|upd=December 1, 2012		\|upd=December 3, 2012
	\|v=1.00}}		\|v=1.01}}
	<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of an operator (i.e., the p-norm of the vector of its k largest singular values). It works with both full and sparse matrices.		<tt>'''kpNorm'''</tt> is a [[List of functions\|function]] that computes the [[(k,p)-norm]] of an operator (i.e., the p-norm of the vector of its k largest singular values). It works with both full and sparse matrices.

Nathaniel: Created page with "{{Function |name=kpNorm |desc=Computes the (k,p)-norm of an operator |rel=KyFanNorm
SchattenNorm
TraceNorm |upd=December 1, 2012 |v=1.00}} `'''kpN..."`

2012-12-01T21:55:20Z Created page with "{{Function |name=kpNorm |desc=Computes the (k,p)-norm of an operator |rel=KyFanNorm<br />SchattenNorm<br />TraceNorm |upd=December 1, 2012 |v=1.00}} <tt>'''kpN..." New page {{Function |name=kpNorm |desc=Computes the [[(k,p)-norm]] of an operator |rel=[[KyFanNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]] |upd=December 1, 2012 |v=1.00}} <tt>'''kpNorm'''</tt> is a [[List of functions|function]] that computes the [[(k,p)-norm]] of an operator (i.e., the p-norm of the vector of its k largest singular values). It works with both full and sparse matrices. ==Syntax== * <tt>NRM = kpNorm(X,K,P)</tt> ==Argument descriptions== * <tt>X</tt>: An operator to have its (<tt>K</tt>,<tt>P</tt>)-norm computed. * <tt>K</tt>: A positive integer. * <tt>P</tt>: A real number ≥ 1, or <tt>Inf</tt>. ==Examples== ===Generalizes the operator, trace, Ky Fan, and Schatten norms=== The (<tt>K</tt>,<tt>P</tt>)-norm is simply the usual [[operator norm]] when <tt>K = 1</tt> or <tt>P = Inf</tt>: <pre> >> X = rand(3); >> [norm(X), kpNorm(X,1,Inf), kpNorm(X,2,Inf), kpNorm(X,3,Inf), kpNorm(X,1,5)] ans = 1.0673 1.0673 1.0673 1.0673 1.0673 </pre> When <tt>P = 1</tt> and <tt>K</tt> is the size of <tt>X</tt>, this norm reduces to the [[trace norm]]: <pre<noinclude></noinclude>> >> [kpNorm(X,3,1), [[TraceNorm|TraceNorm(X)]]] ans = 1.6482 1.6482 </pre<noinclude></noinclude>> More generally, when <tt>P = 1</tt> this norm reduces to the [[Ky Fan norm|Ky Fan <tt>K</tt>-norm]]: <pre<noinclude></noinclude>> >> [kpNorm(X,2,1), [[KyFanNorm|KyFanNorm(X,2)]]] ans = 1.5816 1.5816 </pre<noinclude></noinclude>> Similarly, when <tt>K = min(size(X))</tt> this norm reduces to the [[Schatten norm|Schatten <tt>P</tt>-norm]]: <pre<noinclude></noinclude>> >> [kpNorm(X,3,4), [[SchattenNorm|SchattenNorm(X,4)]]] ans = 1.0814 1.0814 </pre<noinclude></noinclude>>