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
AXe == 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.