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 = A*X;
% 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”.

Why? Please help.