Jump to navigation Jump to search
Computes the dual of the (k,p)-norm of a vector or matrix

Other toolboxes required none
Related functions kpNorm
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 \[\|x\|_{(k,p)} := \left(\sum_{j=1}^k \big|x_i^\downarrow\big|^p \right)^{1/p},\] 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.


  • 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.


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   77.0015

Similarly, 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 =


>> kpNormDual(X,3,2)

ans =


>> [kpNormDual(X,4,2), norm(X,'fro')]

ans =

   72.6498   72.6498

Source code

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


  1. G.S. Mudholkar and M. Freimer. A structure theorem for the polars of unitarily invariant norms. Proc. Amer. Math. Soc., 95:331–337, 1985.