The Fiedler Vector of a Laplacian Tensor for Hypergraph Partitioning

Based on recent advances in spectral hypergraph theory [L. Qi and Z. Luo, Tensor Anaysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017], we explore the Fiedler vector of an even-uniform hypergraph, which is the Z-eigenvector associated with the second smallest Z-eigenvalue of a normalized Laplacian tensor arising from the hypergraph. Then, we develop a novel tensor-based spectral method for partitioning vertices of the hypergraph. For this purpose, we extend the normalized Laplacian matrix of a simple graph to the normalized Laplacian tensor of an even-uniform hypergraph. The corresponding Fiedler vector is related to the Cheeger constant of the hypergraph. Then, we establish a feasible optimization algorithm to compute the Fiedler vector according to the normalized Laplacian tensor. The convergence of the proposed algorithm and the probability of obtaining the Fiedler vector of the hypergraph are analyzed theoretically. Finally, preliminary numerical experiments illustrate that the new a...

[1]  Guillermo Ricardo Simari,et al.  Non-commercial Research and Educational Use including without Limitation Use in Instruction at Your Institution, Sending It to Specific Colleagues That You Know, and Providing a Copy to Your Institution's Administrator. All Other Uses, Reproduction and Distribution, including without Limitation Comm , 2022 .

[2]  Liping Chen,et al.  Computing Tensor Eigenvalues via Homotopy Methods , 2015, SIAM J. Matrix Anal. Appl..

[3]  Bo Jiang,et al.  A framework of constraint preserving update schemes for optimization on Stiefel manifold , 2013, Math. Program..

[4]  J. A. Rodŕıguez,et al.  Laplacian Eigenvalues and Partition Problems in Hypergraphs , 2004 .

[5]  Guangliang Chen,et al.  Spectral Curvature Clustering (SCC) , 2009, International Journal of Computer Vision.

[6]  John Lenz,et al.  Spectral Extremal Problems for Hypergraphs , 2013, SIAM J. Discret. Math..

[7]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

[8]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[9]  M. Fiedler.  A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory , 1975 .

[10]  Liqun Qi,et al.  Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph , 2016, SIAM J. Sci. Comput..

[11]  Yair Weiss,et al.  Segmentation using eigenvectors: a unifying view , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[12]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[13]  Ambedkar Dukkipati,et al.  Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques , 2016, J. Mach. Learn. Res..

[14]  Guoyin Li,et al.  The Z‐eigenvalues of a symmetric tensor and its application to spectral hypergraph theory , 2013, Numer. Linear Algebra Appl..

[15]  Jean Ponce,et al.  A Tensor-Based Algorithm for High-Order Graph Matching , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  L. Qi.  H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors , 2013, 1303.2186.

[17]  Venu Madhav Govindu,et al.  A tensor decomposition for geometric grouping and segmentation , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[18]  Yihong Gong,et al.  Unsupervised Image Categorization by Hypergraph Partition , 2011, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  Achi Brandt,et al.  Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver , 2011, SIAM J. Sci. Comput..

[20]  George Karypis,et al.  Multilevel Hypergraph Partitioning , 2003 .

[21]  Pascal Fua,et al.  SLIC Superpixels Compared to State-of-the-Art Superpixel Methods , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[22]  Robert E. Mahony,et al.  Convergence of the Iterates of Descent Methods for Analytic Cost Functions , 2005, SIAM J. Optim..

[23]  Liqun Qi,et al.  The Laplacian of a uniform hypergraph , 2015, J. Comb. Optim..

[24]  Qun Wang,et al.  Computing Extreme Eigenvalues of Large Scale Hankel Tensors , 2016, J. Sci. Comput..

[25]  Changjiang Bu,et al.  Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs , 2016 .

[26]  Liqun Qi,et al.  Algebraic connectivity of an even uniform hypergraph , 2012, J. Comb. Optim..

[27]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[28]  David J. Kriegman,et al.  From Few to Many: Illumination Cone Models for Face Recognition under Variable Lighting and Pose , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  Marcello Pelillo,et al.  A Game-Theoretic Approach to Hypergraph Clustering , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[30]  Ludmil T. Zikatanov,et al.  A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians , 2014, 1412.0565.

[31]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[32]  Adrian S. Lewis,et al.  The [barred L]ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems , 2006, SIAM J. Optim..

[33]  Bora Uçar,et al.  On Two-Dimensional Sparse Matrix Partitioning: Models, Methods, and a Recipe , 2010, SIAM J. Sci. Comput..

[34]  L. Qi,et al.  Tensor Analysis: Spectral Theory and Special Tensors , 2017 .

[35]  Serge J. Belongie,et al.  Higher order learning with graphs , 2006, ICML.

[36]  Dorothea Wagner,et al.  Modeling Hypergraphs by Graphs with the Same Mincut Properties , 1993, Inf. Process. Lett..