# Low rank approximation in CVX

I try to use CVX toolbox to do “low rank approximation” work. The code is as follows:

r = 2; % the rank;
N = 32; % the dimension
M = 32;
a = randn(N,r);
b = randn(M,r);
X = a*b’; % low rank Matrix;

A = rand(20,N);
Y = AX;
% low rank approximation using nuclear norm
cvx_begin
variable Xe(N,M)
minimize norm_nuc(Xe);
subject to
A
Xe == Y;
cvx_end

Then Matlab tells me that

## number of iterations = 8 primal objective value = 5.75866738e+01 dual objective value = 5.75866738e+01 gap := trace(XZ) = 3.73e-08 relative gap = 3.21e-10 actual relative gap = 3.49e-10 rel. primal infeas (scaled problem) = 6.90e-11 rel. dual " " " = 1.50e-12 rel. primal infeas (unscaled problem) = 0.00e+00 rel. dual " " " = 0.00e+00 norm(X), norm(y), norm(Z) = 8.4e+01, 1.6e+00, 4.2e+00 norm(A), norm(b), norm© = 6.0e+01, 1.6e+02, 5.1e+00 Total CPU time (secs) = 1.75 CPU time per iteration = 0.22 termination code = 0 DIMACS: 4.9e-10 0.0e+00 3.8e-12 0.0e+00 3.5e-10 3.2e-10

Status: Solved
Optimal value (cvx_optval): +57.5867

Obviously, CVX works well and the job has been done.
However, the results “Xe” does not equal to the original matrix “X”.