A Sample Procrustes Problem

In this example, we attempt to find a $Y$ which minimizes $\vert\vert AY-YB\vert\vert _F$ for a pair of randomly selected $A$ ($12 \times 12$) and $B$ ($4 \times 4$).

>> !cp examples/procrustes/*.m .
>> randn('state',0);
>> [A,B] = randprob;
>> parameters(A,B);
>> Y0 = guess;
>> [fn,Yn] = sg_min(Y0,'newton','euclidean');
iter    grad            F(Y)              flops         step type
0       2.334988e+01    3.071299e+01         4751       none
  invdgrad: Hessian not positive definite, CG terminating early
1       1.171339e+01    1.376463e+01       365678       Newton step
  invdgrad: Hessian not positive definite, CG terminating early
2       7.843279e+00    7.616381e+00       677599       Newton step
  invdgrad: Hessian not positive definite, CG terminating early
3       5.131680e+00    4.945824e+00       992823       Newton step
  invdgrad: Hessian not positive definite, CG terminating early
4       5.642834e+00    3.512826e+00      1293761       Newton step
  invdgrad: Hessian not positive definite, CG terminating early
5       5.500553e+00    1.721329e+00      1607969       Newton step
  invdgrad: Hessian not positive definite, CG terminating early
6       4.666307e+00    1.192561e+00      1964675       Newton step
  invdgrad: Hessian not positive definite, CG terminating early
7       3.576069e+00    6.850532e-01      2272355       Newton step
  invdgrad: max iterations reached inverting the Hessian by CG
8       1.228119e+00    3.046816e-01      2820625       Newton step
9       4.673779e-02    2.506848e-01      3345266       Newton step
10      6.411668e-04    2.505253e-01      3873582       Newton step
11      1.965463e-06    2.505253e-01      4430245       Newton step
12      1.620267e-06    2.505253e-01      5022629       Newton step

Figure 9.1 shows the convergence curve for this run.

Figure 9.1: Procrustes problem