Developments in structural learning using Ihara coefficients and hypergraph representations

This thesis addresses the problems in structural learning, particularly focusing on structural characterization and matching. To this end, we present an approach named Ihara coefficients, which is capable of characterizing structures of varying order into pattern space related to prime cycles. Furthermore, we develop a matching algorithm for establishing correspondences between structures by conducting dominant cluster analysis (DCA) on a direct product hypergraph (DPH) . In Chapter 3 we describe how to extract characteristics from the Ihara zeta function for the purpose of clustering graphs. The novel contributions of this chapter are twofold. First, we demonstrate how to characterize unweighted graphs in a permutation invariant manner using the polynomial coefficients from the Ihara zeta function, i.e. the Ihara coefficients. Second, we generalize the definition of the Ihara coefficients to edge-weighted graphs. For an unweighted graph, the Ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph. Since the Ihara zeta function has poles which give rise to infinities, the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial. Moreover, the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph. To generalize the representation to edge-weighted graphs we make use of the reduced Bartholdi zeta function. We prove that the computation of the Ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge-weights. We also present a spectral analysis of the Ihara coefficients and indicate their advantages over other graph spectral methods. We apply the proposed graph characterization method to capturing graph-class structure and clustering graphs. Experimental results reveal that the Ihara coefficients are more effective than methods based on Laplacian spectra. In Chapter 4 we aim to seek a compact characterization of nonuniform unweighted hypergraphs for the purposes of clustering. To this end, we develop a polynomial characterization for hypergraphs based on the Ihara zeta function. We investigate the flexibility of the polynomial coefficients for learning relational structures with different relational orders. Furthermore, we develop an efficient method for computing the coefficient set. Our representation for hypergraphs takes into account not only the vertex connections but also the hyperedge cardinalities, and thus can distinguish different relational orders, which is prone to ambiguity if the hypergraph Laplacian is used. In our experimental evaluation, we demonstrate the effectiveness of the proposed characterization for clustering nonuniform unweighted hypergraphs and its advantages over the spectral characterization of the hypergraph Laplacian. In addition to the flexible characterization algorithms developed based on the Ihara coefficients, we present a novel structural matching algorithm in Chapter 5. This algorithm can be used both for pairwise matching and higher order matching. We formulate the problem of high order structural matching by applying dominant cluster analysis (DCA) to a direct product hypergraph (DPH) (essentially the extension of the association graph idea to hypergraphs). For brevity we refer to the resulting algorithm as DPH-DCA. The starting point for our method is to construct aK-uniform direct product hypergraph for the two sets of higher-order features to be matched. Each vertex in the direct product hypergraph represents a potential correspondence and the weight on each hyperedge represents the agreement between twoK-tuples drawn from the two feature sets. Vertices representing correct assignments tend to form a strongly intra-connected cluster, i.e. a dominant cluster. We evaluate the association of each vertex belonging to the dominant cluster by maximizing an objective function which maintains the K-tuple agreements. The potential correspondences with non-zero association weights are more likely to belong to the dominant cluster than the remaining zero-weighted ones. They are thus selected as correct matchings subject to the one-to-one correspondence constraint. Furthermore, we present a