KyFanNorm

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 + 0.0000i

Source code

Click here to view this function's source code on github.