Fig 1.
An example of smooth signal on graph.
The left panel illustrates the mapping a 3D-PET image to a weighted graph that describes the affinity between every pair of voxels, exemplified by the 3D brain image in [6]. The amplitudes of the first few GFT coefficients of the image are displayed in the right panel, where the amplitudes have been divided by the the maximum value among them.
Fig 2.
Simulation results on a 16-vertex random graph with its Laplacian matrix shown in (a). The data included M = 300 signals on the graph for (b)–(f) and M = 20 signals for (g)–(i). Each signal was a linear combination of the first 6 eigenvectors of the graph Laplacian. For M = 20 or 300, the GRM significantly outperforms the compared methods, namely the sample correlation and the graphical Lasso.
Fig 3.
Performance of the graph structure learning methods v.s. the number of samples.
In the left and right panels, we show the NMI and F-score between the estimated graph and the truth, respectively. The error bar indicates the standard deviation of the measurement. For every M, we ran the experiment 100 times by randomly generating the graph structure and the signals.
Table 1.
Participant demographics.
Table 2.
Names of the VOIs used in the brain connectivity learning model.
Fig 4.
Network learning results of the AD group.
Reconstruction results by the GRM (2nd row) from 30 AD subjects compared with those by sample correlation (1st row) and graphical Lasso (3rd row). We plotted the sagittal and coronal views of the thresholded networks in the first two columns, where 144 prominent links are shown for all cases. The colors and sizes of the nodes indicate the associated brain lobes (frontal lobe is cyan; temporal lobe is pink; parietal lobe is blue; occipital lobe is purple) and node degrees, respectively; the thicknesses of the edges encode the connection strengthes. Those figures were generated by BrainNet Viewer (http://www.nitrc.org/projects/bnv/). We used s = 2, β = −0.003 and ρ = 0.002 to simulate (f) and (i), respectively. To ease visualization, we removed the diagonal entries when displaying the graph Laplacian and partial correlation matrices.
Fig 5.
Network learning results of the NC group.
Reconstruction results by the GRM from 40 NC subjects compared with those by sample correlation and graphical Lasso. For both GRM and graphical Lasso, results are obtained by keeping the same parameters as in the previous AD case. More detailed descriptions of the figure generation can be found in the caption of Fig 4.
Fig 6.
ROC curves of PiB-PET data classification.
ROC curves of PiB-PET data classification for the proposed GRM and sample correlation (SC), graphical Lasso (GLasso). When using GRM, we compare the variations of the testing data point on the graphs learned from the training data of the AD group and NC group, respectively. When using SC or GLasso, we compare the Mahalanobis distances or the likelihood of the new data point under the two Gaussian graphical models, accordingly. The GRM significantly outperforms the other two methods.
Table 3.
Performance metrics and parameters of PiB-PET data classification.
Table 4.
Measurements of the weighted graphs obtained by different methods.
Table 5.
Average degree of the weighted networks learned by different approaches.
Fig 7.
Degree distributions of the thresholded networks.
Degree distributions of the thresholded AD and NC networks obtained by different methods. The horizontal and vertical axes correspond to the brain region index and vertex degree, respectively. The magenta lines delineate the four brain lobes: namely, the frontal, parietal, occipital, and temporal lobe (from left to right).
Fig 8.
Comparison between the AD and NC networks learned by GRM.
(a)-(b) correspond to the adjacency matrices of thresholded networks with both 187 edges, respectively; (c) visualizes the degree distribution differences between the AD and NC networks, with respect to the network density. Both the horizontal and vertical axes in the first two diagrams represent the brain region index.
Fig 9.
Adjacency matrices obtained by NMI.
Adjacency matrices obtained by computing the normalized mutual information between observations of every pair of brain regions, for the AD group (left panel) and the NC group (right panel), respectively.