KyFanNorm

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.

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</tt> function:

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.