KpNormDual: Difference between revisions
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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= | |upd=June 24, 2015 | ||
|cvx= | |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> | ||
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. | 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== | ||
Latest revision as of 14:45, 25 May 2016
| kpNormDual | |
| Computes the dual of the (k,p)-norm of a vector or matrix | |
| Other toolboxes required | none |
|---|---|
| Related functions | kpNorm KyFanNorm SchattenNorm TraceNorm |
| Function category | Norms |
| Usable within CVX? | yes (convex) |
kpNormDual is a 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 $|x_1^\downarrow| \geq |x_2^\downarrow| \geq \cdots \geq |x_n^\downarrow|$. 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[1]. This function 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 77.0015Similarly, 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 72.6498Source code
Click here to view this function's source code on github.
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.