Intrinsic Graph Structure Estimation Using Graph Laplacian

A graph is a mathematical representation of a set of variables where some pairs of the variables are connected by edges. Common examples of graphs are railroads, the Internet, and neural networks. It is both theoretically and practically important to estimate the intensity of direct connections between variables. In this study, a problem of estimating the intrinsic graph structure from observed data is considered. The observed data in this study are a matrix with elements representing dependency between nodes in the graph. The dependency represents more than direct connections because it includes influences of various paths. For example, each element of the observed matrix represents a co-occurrence of events at two nodes or a correlation of variables corresponding to two nodes. In this setting, spurious correlations make the estimation of direct connection difficult. To alleviate this difficulty, a digraph Laplacian is used for characterizing a graph. A generative model of this observed matrix is proposed, and a parameter estimation algorithm for the model is also introduced. The notable advantage of the proposed method is its ability to deal with directed graphs, while conventional graph structure estimation methods such as covariance selections are applicable only to undirected graphs. The algorithm is experimentally shown to be able to identify the intrinsic graph structure.

[1]  Xi Luo.  High Dimensional Low Rank and Sparse Covariance Matrix Estimation via Convex Minimization , 2011 .

[2]  Kaspar Riesen,et al.  Recent advances in graph-based pattern recognition with applications in document analysis , 2011, Pattern Recognit..

[3]  John M. Beggs,et al.  A Maximum Entropy Model Applied to Spatial and Temporal Correlations from Cortical Networks In Vitro , 2008, The Journal of Neuroscience.

[4]  Michael J. Berry,et al.  Weak pairwise correlations imply strongly correlated network states in a neural population , 2005, Nature.

[5]  R. Tibshirani,et al.  Sparse inverse covariance estimation with the graphical lasso. , 2008, Biostatistics.

[6]  Donald F. Towsley,et al.  Estimating and sampling graphs with multidimensional random walks , 2010, IMC '10.

[7]  Sreenivas Gollapudi,et al.  Estimating PageRank on graph streams , 2008, PODS.

[8]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

[9]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[10]  Hichem Sahbi,et al.  Manifold learning using robust Graph Laplacian for interactive image search , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[11]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[12]  Katya Scheinberg,et al.  IBM Research Report SINCO - A Greedy Coordinate Ascent Method for Sparse Inverse Covariance Selection Problem , 2009 .

[13]  Gilles Villard,et al.  Fast Parallel Computation of the Jordan Normal Form of Matrices , 1996, Parallel Process. Lett..

[14]  M. Kendall,et al.  Rank Correlation Methods , 1949 .

[15]  Luciano Lopez,et al.  The Cayley transform in the numerical solution of unitary differential systems , 1998, Adv. Comput. Math..

[16]  Alexandre d'Aspremont,et al.  Model Selection Through Sparse Max Likelihood Estimation Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data , 2022 .

[17]  N. Higham.  Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics) , 2008 .

[18]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[19]  John D. Lafferty,et al.  Diffusion Kernels on Graphs and Other Discrete Input Spaces , 2002, ICML.

[20]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[21]  Pradeep Ravikumar,et al.  Sparse inverse covariance matrix estimation using quadratic approximation , 2011, MLSLP.

[22]  Zhi-Li Zhang,et al.  Random Walks on Digraphs, the Generalized Digraph Laplacian and the Degree of Asymmetry , 2010, WAW.

[23]  Takashi Washio,et al.  State of the art of graph-based data mining , 2003, SKDD.

[24]  Aapo Hyvärinen,et al.  A Linear Non-Gaussian Acyclic Model for Causal Discovery , 2006, J. Mach. Learn. Res..

[25]  Jeffrey C. Lagarias,et al.  Convergence Properties of the Nelder-Mead Simplex Method in Low Dimensions , 1998, SIAM J. Optim..

[26]  Shun-ichi Amari,et al.  Information-Geometric Measures as Robust Estimators of Connection Strengths and External Inputs , 2009, Neural Computation.

[27]  Mikhail Belkin,et al.  Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering , 2001, NIPS.

[28]  C. F. Kossack,et al.  Rank Correlation Methods , 1949 .

[29]  Larry A. Wasserman,et al.  Stability Approach to Regularization Selection (StARS) for High Dimensional Graphical Models , 2010, NIPS.