Large Scale Spectral Clustering Using Approximate Commute Time Embedding

Spectral clustering is a novel clustering method which can detect complex shapes of data clusters. However, it requires the eigen decomposition of the graph Laplacian matrix, which is proportion to $O(n^3)$ and thus is not suitable for large scale systems. Recently, many methods have been proposed to accelerate the computational time of spectral clustering. These approximate methods usually involve sampling techniques by which a lot information of the original data may be lost. In this work, we propose a fast and accurate spectral clustering approach using an approximate commute time embedding, which is similar to the spectral embedding. The method does not require using any sampling technique and computing any eigenvector at all. Instead it uses random projection and a linear time solver to find the approximate embedding. The experiments in several synthetic and real datasets show that the proposed approach has better clustering quality and is faster than the state-of-the-art approximate spectral clustering methods

Learning from Multiple Graphs using a Sigmoid Kernel

International audienceThis paper studies the problem of learning from a set of input graphs, each of them representing a different relation over the same set of nodes. Our goal is to merge those input graphs by embedding them into an Euclidean space related to the commute time distance in the original graphs. This is done with the help of a small number of labeled nodes. Our algorithm output a combined kernel that can be used for different graph learning tasks. We consider two combination methods: the (classical) linear combination and the sigmoid combination. We compare the combination methods on node classification tasks using different semi-supervised graph learning algorithms. We note that the sigmoid combination method exhibits very positive results

3D Shape Registration Using Spectral Graph Embedding and Probabilistic Matching

International audienceIn this book chapter we address the problem of 3D shape registration and we propose a novel technique based on spectral graph theory and probabilistic matching. Recent advancement in shape acquisition technology has led to the capture of large amounts of 3D data. Existing real-time multi-camera 3D acquisition methods provide a frame-wise reliable visual-hull or mesh representations for real 3D animation sequences The task of 3D shape analysis involves tracking, recognition, registration, etc. Analyzing 3D data in a single framework is still a challenging task considering the large variability of the data gathered with different acquisition devices. 3D shape registration is one such challenging shape analysis task. The main contribution of this chapter is to extend the spectral graph matching methods to very large graphs by combining spectral graph matching with Laplacian embedding. Since the embedded representation of a graph is obtained by dimensionality reduction we claim that the existing spectral-based methods are not easily applicable. We discuss solutions for the exact and inexact graph isomorphism problems and recall the main spectral properties of the combinatorial graph Laplacian; We provide a novel analysis of the commute-time embedding that allows us to interpret the latter in terms of the PCA of a graph, and to select the appropriate dimension of the associated embedded metric space; We derive a unit hyper-sphere normalization for the commute-time embedding that allows us to register two shapes with different samplings; We propose a novel method to find the eigenvalue-eigenvector ordering and the eigenvector sign using the eigensignature (histogram) which is invariant to the isometric shape deformations and fits well in the spectral graph matching framework, and we present a probabilistic shape matching formulation using an expectation maximization point registration algorithm which alternates between aligning the eigenbases and finding a vertex-to-vertex assignment

Graph similarity through entropic manifold alignment

In this paper we decouple the problem of measuring graph similarity into two sequential steps. The first step is the linearization of the quadratic assignment problem (QAP) in a low-dimensional space, given by the embedding trick. The second step is the evaluation of an information-theoretic distributional measure, which relies on deformable manifold alignment. The proposed measure is a normalized conditional entropy, which induces a positive definite kernel when symmetrized. We use bypass entropy estimation methods to compute an approximation of the normalized conditional entropy. Our approach, which is purely topological (i.e., it does not rely on node or edge attributes although it can potentially accommodate them as additional sources of information) is competitive with state-of-the-art graph matching algorithms as sources of correspondence-based graph similarity, but its complexity is linear instead of cubic (although the complexity of the similarity measure is quadratic). We also determine that the best embedding strategy for graph similarity is provided by commute time embedding, and we conjecture that this is related to its inversibility property, since the inverse of the embeddings obtained using our method can be used as a generative sampler of graph structure.The work of the first and third authors was supported by the projects TIN2012-32839 and TIN2015-69077-P of the Spanish Government. The work of the second author was supported by a Royal Society Wolfson Research Merit Award

Weighted Spectral Embedding of Graphs

We present a novel spectral embedding of graphs that incorporates weights assigned to the nodes, quantifying their relative importance. This spectral embedding is based on the first eigenvectors of some properly normalized version of the Laplacian. We prove that these eigenvectors correspond to the configurations of lowest energy of an equivalent physical system, either mechanical or electrical, in which the weight of each node can be interpreted as its mass or its capacitance, respectively. Experiments on a real dataset illustrate the impact of weighting on the embedding