# Computing or bounding a modified Ky Fan norm

The Ky Fan k-norm of a matrix X \in M_n is defined as follows:

\|X\|_k := \sum_{i=1}^k \sigma_i^\downarrow(X)

(i.e., the sum of the k largest singular values of X). In CVX, there is a simple way to optimize over the set of matrices with bounded Ky Fan k-norm. For example, if we wanted to optimize over the set of matrices with Ky Fan k-norm no larger than 1, we could set one of the constraints in CVX as follows:

lambda_sum_largest([zeros(n),X;X',zeros(n)],k) <= 1;


My question is whether or not there is a similarly simple way to implement the following norm in CVX:

\|X\|_{(k,2)} := \sqrt{\sum_{i=1}^k (\sigma_i^\downarrow(X))^2}

That is, I am interested in the 2-norm of the k largest singular values instead of the 1-norm of the k largest singular values.

I am not aware of a way to do this, no. You would need to be able to express the epigraph of this norm using semidefinite constraints.

For completeness’ sake, this question can be solved by using Madeleine Udell’s wonderful answer here elsewhere on these forums:

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