Experiments and Results

We experiment with the wave kernel embedding as a graph characterization for the purposes of graph-matching. We represent the graphs under study using sets of coordinate vectors corresponding to the embedded node position, and compute the similarity of the sets resulting from different graphs using the robust modified Hausdorff distance.

In our experiments we aim to investigate whether the edge-based wave kernel embedding can be used as a graph characterization, for gauging the similarity of graphs, and hence clustering them. To commence, we compute the eigensystem of the edge-based Laplacian from the eigensystem of the normalized adjacency matrix, and hence compute the edge-based Laplacian matrix introduced in Sec-tion 5.2.2. The edge-based wave kernel then is computed as described in SecSec-tion 5.2.1 with the values of t = 10.0, 1.0, 0.1 and 0.01. From the wave-kernel we compute the embedding coordinate matrix, whose columns are the coordinates of the embedded nodes in a pseudo-Euclidean space. Finally, we project the co-ordinate vectors onto a pseudo-Euclidean space with low dimension using

t=10 t=1.0 t=0.1 t=0.01 Houses data 0.2333 0.0000 0.0333 0.1000 COIL data 0.3333 0.3333 0.3333 0.7000 Table 5.1: A rand index vs. t for the York model house dataset

the orthonormal basis as shown in Section 5.3. With the vector representations residing in a low dimension space we construct the distance matrices between the thirty different graphs using both the classical and modified Hausdorff dis-tance 3.4.1. Finally, we subject the disdis-tance matrices to multidimensional scal-ing MDS (Cox & Cox, 1994) to embed them into a 3D space. Here each graph is represented by a single point. Figure 5.2 shows the results obtained using the modified Hausdorff distance. The subfigures are ordered from left to right (up to down), using t = 10.0, 1.0, 0.1 and 0.01 in the wave kernel. We have also investigated the COIL data, and the results are shown in Figure 5.4.

We commence by introducing the results obtained when experimenting with the York model house database. Where the CMU model house sequence is repre-sented as a red circle and each graph of the MOVI model house sequence is rep-resented as blue star while each graph of the Swiss chalet model house sequence is represented as a green cross. To commence, we show in Figures 5.1 and 5.2 the results when using the Hausdorff distance (HD) and the modified Hausdorff distance (MHD) to measure the (dis)similarity between pairs of graphs, respec-tively. The subfigures are ordered from left to right, top to bottom using the heat kernel embedding with the values t = 10.0, 1.0, 0.1 and 0.01 respectively. With the same order, Figures 5.3 and 5.4, give the results obtained when using COIL-20 dataset where each graph of the sequence of the first object is represented as a red circle and each graph of the sequence of the second object is represented as blue star while each graph of the sequence of the third object is represented as a green cross..

To investigate the results in more details table 4.1 shows the rand index for the distance as a function of t. This index is computed as explained in Section 3.4.5.

Although, the wave kernel gives a reasonable separation of the objects into

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Figure 5.1: MDS embedding obtained when using HD for the Wave Kernel em-bedding for the houses data.

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Figure 5.2: MDS embedding obtained when using MHD for the Wave Kernel embedding for the houses data.

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Figure 5.3: MDS embedding obtained when using HD for the Wave Kernel em-bedding for the COIL data .

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Figure 5.4: MDS embedding obtained when using MHD for the Wave Kernel embedding for the COIL data .

distinct clusters particularly for value of t equal to 1, the experimental study shows that it needs further development; for that reason is we suggest in Ap-pendix I to generalize the wave kernel framework in higher dimensional space.

However, we can figure out a number of conclusions to be drawn from the plots.

For instance, the sequence of the second object of the COIL dataset is clustered along a straight line for all values of t, while the other two sequence are embed-ded in a less compact cluster. Whilst, the York model house dataset gives a more obvious clusters than those of the COIL dataset. Unlike the situation when using the heat kernel embedding where we project the data into a positive space, in the wave kernel case we try to preserve the geometry of the original data.

5.5 Conclusion

In this chapter we have established a procedure to embed the nodes of a graph into a pseudo-Riemannian manifold based on the wave kernel, which is the solu-tion of an edge-based wave equasolu-tion. Under the embedding, each edge became a geodesic on the manifold. The eigensystem of the wave-kernel was determined by the eigenvalues and the eigenfunctions of the normalized adjacency matrix.

By factorizing the Gram-matrix for the wave-kernel, we determine the embed-ding co-ordinates for nodes under the wave-kernel. We investigated the utility of this new embedding as a means of gauging the similarity of graphs. We ex-perimented on sets of graphs representing the proximity of image features in different views of different objects from two datasets (the York Model House and COIL datasets). And by applying multidimensional scaling to the similar-ity matrix we demonstrated that the proposed graph representation is capable of clustering different views of the same object together.