Table 3.
Comparison between the correlations obtained by the eigenvector method ρeig, by the SDP method ρsdp and by the least squares method ρlsqr for different values of p (small world graph on S2, n = 200, ε = 0.3, m ≈ 3000). The SDP tends to find low-rank matrices despite the fact that the rank-one constraint on Θ is not included in the SDP. The rightmost column gives the rank of the Θ matrices that were found by the SDP. To solve the SDP (8)–(10) we used SDPLR, a package for solving large-scale SDP problems [5]. The least squares solution was obtained using MATLAB’s lsqr function. As expected, the least squares method yields poor correlations compared to the eigenvector and the SDP methods.
| p | ρlsqr | ρeig | ρsdp | rank Θ |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 0.7 | 0.787 | 0.977 | 0.986 | 1 |
| 0.4 | 0.046 | 0.839 | 0.893 | 3 |
| 0.3 | 0.103 | 0.560 | 0.767 | 3 |
| 0.2 | 0.227 | 0.314 | 0.308 | 4 |
| 0.15 | 0.091 | 0.114 | 0.102 | 5 |