KpNorm: Difference between revisions
Jump to navigation
Jump to search
mNo edit summary |
Updated to v1.02 |
||
| Line 1: | Line 1: | ||
{{Function | {{Function | ||
|name=kpNorm | |name=kpNorm | ||
|desc=Computes the [[(k,p)-norm]] of | |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= | |upd=March 9, 2013 | ||
|v=1. | |v=1.02}} | ||
<tt>'''kpNorm'''</tt> is a [[List of functions|function]] that computes the [[(k,p)-norm]] of | <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== | ||
| Line 11: | Line 11: | ||
==Argument descriptions== | ==Argument descriptions== | ||
* <tt>X</tt>: | * <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>. | ||
| Line 17: | Line 17: | ||
==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); | ||
Revision as of 21:40, 25 June 2013
| kpNorm | |
| Computes the (k,p)-norm of a vector or matrix | |
| Other toolboxes required | none |
|---|---|
| Related functions | kpNormDual KyFanNorm SchattenNorm TraceNorm |
kpNorm is a 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
- NRM = kpNorm(X,K,P)
Argument descriptions
- X: A vector or matrix to have its (K,P)-norm computed.
- K: A positive integer.
- P: A real number ≥ 1, or Inf.
Examples
Generalizes the operator, trace, Ky Fan, and Schatten norms
The (K,P)-norm of a matrix is simply the usual operator norm when K = 1 or P = Inf:
>> 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
When P = 1 and K is the size of X, this norm reduces to the trace norm:
>> [kpNorm(X,3,1), TraceNorm(X)] ans = 1.6482 1.6482
More generally, when P = 1 this norm reduces to the Ky Fan K-norm:
>> [kpNorm(X,2,1), KyFanNorm(X,2)] ans = 1.5816 1.5816
Similarly, when K = min(size(X)) this norm reduces to the Schatten P-norm:
>> [kpNorm(X,3,4), SchattenNorm(X,4)] ans = 1.0814 1.0814