KpNormDual: Difference between revisions
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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..." |
Updated to v1.02 |
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{{Function | {{Function | ||
|name=kpNormDual | |name=kpNormDual | ||
|desc=Computes the [[dual of the (k,p)-norm]] of | |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= | |upd=April 3, 2013 | ||
|v=1. | |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>: | * <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]; | ||
Revision as of 21:42, 25 June 2013
| kpNormDual | |
| Computes the dual of the (k,p)-norm of a vector or matrix | |
| Other toolboxes required | kpNorm |
|---|---|
| Related functions | KyFanNorm SchattenNorm TraceNorm |
kpNormDual is a function that computes the dual of the (k,p)-norm[1] of a vector or matrix. It works with both full and sparse vectors and matrices.
Syntax
- NRM = kpNormDual(X,K,P)
Argument descriptions
- X: A vector or matrix to have its norm computed.
- K: A positive integer.
- P: A real number ≥ 1, or Inf.
Examples
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):
>> X = [1 1 1 1;1 2 3 4;1 4 9 16;1 8 27 64]; >> [kpNormDual(X,1,1), TraceNorm(X)] ans = 77.0015 77.0015
Similarly, if K = min(size(X)) and P = 2 then kpNorm(X,K,P) is the Frobenius norm, which is its own dual. Thus kpNormDual(X,K,2) decreases from the trace norm of X to its Frobenius norm as K increases:
>> [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
References
- ↑ G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. Proc. Amer. Math. Soc., 95:331–337, 1985.