Difference between revisions of "KyFanNorm"
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(Created page with "{{Function |name=KyFanNorm |desc=Computes the Ky Fan k-norm of an operator |rel=kpNorm<br />SchattenNorm<br />TraceNorm |upd=December 1, 2012 |v=1....") |
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{{Function | {{Function | ||
|name=KyFanNorm | |name=KyFanNorm | ||
| − | |desc=Computes the | + | |desc=Computes the Ky Fan norm|Ky Fan k-norm of an operator |
|rel=[[kpNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]] | |rel=[[kpNorm]]<br />[[SchattenNorm]]<br />[[TraceNorm]] | ||
| + | |cat=[[List of functions#Norms|Norms]] | ||
|upd=December 1, 2012 | |upd=December 1, 2012 | ||
| − | | | + | |cvx=yes (convex)}} |
| − | <tt>'''KyFanNorm'''</tt> is a [[List of functions|function]] that computes the | + | <tt>'''KyFanNorm'''</tt> is a [[List of functions|function]] that computes the Ky Fan k-norm of an operator (i.e., the sum of its k largest singular values): |
| + | : <math>\|X\|_{(k)} := \sum_{j=1}^k \sigma_j^{\downarrow}(X).</math> | ||
| + | This function works with both full and sparse matrices, and can be used in the objective function or constraints of a CVX optimization problem. | ||
==Syntax== | ==Syntax== | ||
| Line 16: | Line 19: | ||
==Examples== | ==Examples== | ||
===Equals the operator norm when <tt>K = 1</tt> (but faster!)=== | ===Equals the operator norm when <tt>K = 1</tt> (but faster!)=== | ||
| − | The Ky Fan 1-norm is just the operator norm, which is implemented in MATLAB by the <tt>[http://www.mathworks.com/help/ | + | The Ky Fan 1-norm is just the operator norm, which is implemented in MATLAB by the <tt>[http://www.mathworks.com/help/matlab/ref/norm.html norm]</tt> function. However, the <tt>KyFanNorm</tt> function is typically much faster than the <tt>norm</tt> function: |
| − | < | + | <syntaxhighlight> |
>> X = rand(2500); | >> X = rand(2500); | ||
>> tic; KyFanNorm(X,1) | >> tic; KyFanNorm(X,1) | ||
| Line 27: | Line 30: | ||
Elapsed time is 0.861377 seconds. | Elapsed time is 0.861377 seconds. | ||
| + | |||
>> tic; norm(X) | >> tic; norm(X) | ||
toc | toc | ||
| Line 35: | Line 39: | ||
Elapsed time is 4.288788 seconds. | Elapsed time is 4.288788 seconds. | ||
| − | </ | + | </syntaxhighlight> |
| + | |||
| + | ===Can be used with CVX=== | ||
| + | This function can be used directly in constraints or the objective function of CVX problems. The following code snippet computes the minimum value of $\mathrm{Tr}(S\rho)$ over all density matrices $\rho$ satisfying $\|\rho\|_{(2)} \leq 3/4$, where $S$ is the [[SwapOperator|swap operator]]. | ||
| + | <syntaxhighlight> | ||
| + | >> cvx_begin sdp quiet | ||
| + | variable rho(4,4) hermitian; | ||
| + | minimize trace(rho*SwapOperator(2)); | ||
| + | subject to | ||
| + | trace(rho) == 1; | ||
| + | rho >= 0; | ||
| + | KyFanNorm(rho,2) <= 3/4; | ||
| + | cvx_end | ||
| + | cvx_optval | ||
| + | |||
| + | cvx_optval = | ||
| + | |||
| + | -0.2500 | ||
| + | |||
| + | >> rho | ||
| + | |||
| + | rho = | ||
| + | |||
| + | 0.1250 + 0.0000i -0.0000 + 0.0000i -0.0000 + 0.0000i -0.0000 + 0.0000i | ||
| + | -0.0000 - 0.0000i 0.3750 + 0.0000i -0.2500 + 0.0000i -0.0000 + 0.0000i | ||
| + | -0.0000 - 0.0000i -0.2500 - 0.0000i 0.3750 + 0.0000i -0.0000 + 0.0000i | ||
| + | -0.0000 - 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i 0.1250 + 0.0000i | ||
| + | </syntaxhighlight> | ||
| + | |||
| + | {{SourceCode|name=KyFanNorm}} | ||
Latest revision as of 16:39, 24 December 2014
| KyFanNorm | |
| Computes the Ky Fan norm | |
| Other toolboxes required | none |
|---|---|
| Related functions | kpNorm SchattenNorm TraceNorm |
| Function category | Norms |
| Usable within CVX? | yes (convex) |
KyFanNorm is a function that computes the Ky Fan k-norm of an operator (i.e., the sum of its k largest singular values): \[\|X\|_{(k)} := \sum_{j=1}^k \sigma_j^{\downarrow}(X).\] This function works with both full and sparse matrices, and can be used in the objective function or constraints of a CVX optimization problem.
Syntax
- NRM = KyFanNorm(X,K)
Argument descriptions
- X: An operator to have its Ky Fan K-norm computed.
- K: A positive integer.
Examples
Equals the operator norm when K = 1 (but faster!)
The Ky Fan 1-norm is just the operator norm, which is implemented in MATLAB by the norm function. However, the KyFanNorm function is typically much faster than the norm function:
>> X = rand(2500);
>> tic; KyFanNorm(X,1)
toc
ans =
1250.2
Elapsed time is 0.861377 seconds.
>> tic; norm(X)
toc
ans =
1250.2
Elapsed time is 4.288788 seconds.Can be used with CVX
This function can be used directly in constraints or the objective function of CVX problems. The following code snippet computes the minimum value of $\mathrm{Tr}(S\rho)$ over all density matrices $\rho$ satisfying $\|\rho\|_{(2)} \leq 3/4$, where $S$ is the swap operator.
>> cvx_begin sdp quiet
variable rho(4,4) hermitian;
minimize trace(rho*SwapOperator(2));
subject to
trace(rho) == 1;
rho >= 0;
KyFanNorm(rho,2) <= 3/4;
cvx_end
cvx_optval
cvx_optval =
-0.2500
>> rho
rho =
0.1250 + 0.0000i -0.0000 + 0.0000i -0.0000 + 0.0000i -0.0000 + 0.0000i
-0.0000 - 0.0000i 0.3750 + 0.0000i -0.2500 + 0.0000i -0.0000 + 0.0000i
-0.0000 - 0.0000i -0.2500 - 0.0000i 0.3750 + 0.0000i -0.0000 + 0.0000i
-0.0000 - 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i 0.1250 + 0.0000iSource code
Click here to view this function's source code on github.