Hierarchical graph Laplacian eigen transforms

c **** Japan Society for Industrial and Applied Mathematics JSIAM Letters Vol.** (****) pp.1– Hierarchical graph Laplacian eigen transforms Jeff Irion 1 and Naoki Saito 1 Department of Mathematics, University of California, Davis, CA 95616 USA E-mail jlirion@math.ucdavis.edu, saito@math.ucdavis.edu Received Abstract We describe a new transform that generates a dictionary of bases for handling data on a graph by combining recursive partitioning of the graph and the Laplacian eigenvectors of each subgraph. Similar to the wavelet packet and local cosine dictionaries for regularly sampled signals, this dictionary of bases on the graph allows one to select an orthonormal basis that is most suitable to one’s task at hand using a best-basis type algorithm. We also describe a few related transforms including a version of the Haar wavelet transform on a graph, each of which may be useful in its own right. Keywords graph Laplacian eigenvectors, Fiedler vectors, spectral graph partitioning, a dic- tionary of orthonormal bases, wavelet-like transforms on graphs Research Activity Group Wavelet Analysis 1. Introduction For signal processing on regular domains, wavelets have both a well-developed theory and a proven track record of success. Accordingly, efforts have been made to extend classical wavelets and wavelet techniques to the ever-expanding realm of data on graphs. Such datasets include structural/morphological data (e.g., tracings of neuronal dendrites), traffic and transportation data, and social networks. The motivation for developing these so- called “second generation wavelets” is simple: to deter- mine whether they afford the same advantages offered by classical wavelets for approximation/compression, de- noising, and classification in this more general setting. However, a key difficulty in extending wavelets to graphs is that we lack the notion of “frequency” in gen- eral, i.e., we cannot apply the Littlewood-Paley theory directly. Therefore, a common strategy has been to de- velop wavelet-like transforms rather than true general- izations of classical wavelets; see e.g., [1–9]. In this ar- ticle, we propose a new redundant transform for data on graphs, along with two variations, and then show the basis vectors computed on a particular graph. 2. Definitions and notation Let G be an undirected connected graph, let V (G) and E(G) denote its vertices and edges, respectively, and let N := |V (G)|. Let W (G) = (W ij ) ∈ R N ×N be the sym- metric weight matrix of G, where W ij denotes the edge weight between vertices i and j. In an unweighted (i.e., combinatorial) graph, W ij is either 0 or 1, depending on whether there is an edge between the two vertices. By contrast, in a weighted graph, W ij indicates the proxim- ity of vertices i, j or affinity of information measured at i, j. Let f = (f (1), . . . , f (N )) T ∈ R N be a data vector, where f (i) is the value measured at the vertex i of the graph. Let 1 := (1, . . . , 1) T ∈ R N . A standard technique for working with data on a graph is to utilize the eigenvectors of the Laplacian ma- trix of the graph, which is defined as L(G) := D(G) − W (G), where D(G) P = diag(d i ) is the (diagonal) degree matrix with d i := j W ij . Alternatively, we may use the random-walk normalized Laplacian, which is defined as L rw (G) := D(G) −1 L(G) = I −D(G) −1 W (G). The eigen- vectors of both L(G) and L rw (G) form a basis of R N and can thus be used for representation, approximation, and analysis of data on G. The simple path graph P N con- sisting of N vertices provides an important insight for the development of our new transform. As pointed out in [10], the eigenvectors of L(P N ) are nothing but the Dis- crete Cosine Transform (DCT) Type II, which are used in the JPEG image compression standard. In general, it is difficult to know the essential support of the Lapla- cian eigenvectors a priori, which strongly depends on the structure of the graph: sometimes they are completely global, like those of P N , while the other times they may be quite localized, as shown in [10]. Hence, it is worth controlling the support of the eigenvectors explicitly. 3. Hierarchical graph Laplacian eigen transform (HGLET) We now introduce our Hierarchical Graph Laplacian Eigen Transform (HGLET). First, we compute the com- plete set of eigenvectors of L(G): φ 00,0 , φ 00,1 , . . . , φ 00,N −1 with corresponding eigenvalues 0 = λ 00,0 < λ 00,1 ≤ · · · ≤ λ 00,N −1 . As this is a multiscale transform, we adopt the notation (λ jk,l , φ jk,l ) for the eigenpairs, with j denoting the level (or depth) of the partition, k denoting the re- gion number on level j, and l indexing the eigenvectors for region k on level j. Then we partition the graph into two disjoint subgraphs (or regions) according to the sign of the Fiedler vector, φ 00,1 . Partitioning the graph in this manner is supported by the theory discussed in [11]. Fur- thermore, the Fiedler vector of L(G) (or L rw (G)) is the solution of the relaxed RatioCut (or Normalized Cut)