ABSTRACT
A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.
It has been conjectured that an analogous characterization holds for higher multiplicities, i.e., there are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into Rk, and then apply geometric considerations to the embedding.
We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n/k and λk, the kth smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O(√(λk log k)). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result [LRTV12]. The √(log k) bound is tight, up to constant factors, for the "noisy hypercube" graphs.
Supplemental Material
- Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. In FOCS, pages 563--572, Washington, DC, USA, 2010. IEEE Computer Society. Google ScholarDigital Library
- Bengt Aspvall and John R. Gilbert. Graph coloring using eigenvalue decomposition. Technical report, Ithaca, NY, USA, 1983. Google ScholarDigital Library
- Noga Alon and Nabil Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing, 26:1733--1748, 1997. Google ScholarDigital Library
- N Alon. Eigenvalues and expanders. Combinatorica, 6:83--96, January 1986. Google ScholarDigital Library
- N. Alon and V. Milman. Isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory, Series B, 38(1):73--88, feb 1985.Google ScholarCross Ref
- William Beckner. Inequalities in Fourier analysis. Ann. of Math. (2), 102(1):159--182, 1975.Google ScholarCross Ref
- Aline Bonami. Étude des coefficients de Fourier des fonctions de Lp(G). Ann. Inst. Fourier (Grenoble), 20(fasc. 2):335--402 (1971), 1970.Google Scholar
- S. Brin and L. Page. The anatomy of a large-scale hypertextual web search engine. In Proceedings of the seventh International Wide Web Conference, 1998. Google ScholarDigital Library
- 8}CCGGP98M. Charikar, C. Chekuri, A. Goel, S. Guha, and S. Plotkin. Approximating a finite metric by a small number of tree metrics. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, 1998. Google ScholarDigital Library
- M. Charikar, C. Chekuri, A. Goel, and S. Guha. Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median. In 30th Annual ACM Symposium on Theory of Computing, pages 114--123. ACM, New York, 1998. Google ScholarDigital Library
- F. R. K. Chung. Laplacians of graphs and Cheeger's inequalities. In Combinatorics, Paul Erdos is eighty, Vol. 2 (Keszthely, 1993), volume 2 of Bolyai Soc. Math. Stud., pages 157--172. János Bolyai Math. Soc., Budapest, 1996.Google Scholar
- Fan R. K. Chung. Spectral graph theory, volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.Google Scholar
- J. Fakcharoenphol and K. Talwar. An improved decomposition theorem for graphs excluding a fixed minor. In Proceedings of 6th Workshop on Approximation, Randomization, and Combinatorial Optimization, volume 2764 of Lecture Notes in Computer Science, pages 36--46. Springer, 2003.Google ScholarCross Ref
- Anupam Gupta, Robert Krauthgamer, and James R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In 44th Symposium on Foundations of Computer Science, pages 534--543, 2003. Google ScholarDigital Library
- Jon M. Kleinberg. Authoritative sources in a hyperlinked environment. Journal of the ACM, 46:668--677, 1999. Google ScholarDigital Library
- J. Kelner, J. R. Lee, G. Price, and S.-H. Teng. Metric uniformization and spectral bounds for graphs. Geom. Funct. Anal., 21(5):1117--1143, 2011.Google ScholarCross Ref
- Philip N. Klein, Serge A. Plotkin, and Satish Rao. Excluded minors, network decomposition, and multicommodity flow. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 682--690, 1993. Google ScholarDigital Library
- James R. Lee and Assaf Naor. Extending Lipschitz functions via random metric partitions. Invent. Math., 160(1):59--95, 2005.Google ScholarCross Ref
- Anand Louis, Prasad Raghavendra, Prasad Tetali, and Santosh Vempala. Algorithmic extensions of Cheeger's inequality to higher eigenvalues and partitions. In APPROX-RANDOM, pages 315--326, 2011. Google ScholarDigital Library
- Anand Louis, Prasad Raghavendra, Prasad Tetali, and Santosh Vempala. Many sparse cuts via higher eigenvalues. In STOC, 2012. Google ScholarDigital Library
- J. R. Lee and A. Sidiropoulos. Genus and the geometry of the cut graph. In Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 193--201, 2010. Google ScholarDigital Library
- Michel Ledoux and Michel Talagrand. Probability in Banach spaces. Classics in Mathematics. Springer-Verlag, Berlin, 2011. Isoperimetry and processes, Reprint of the 1991 edition.Google Scholar
- J. Matousek. Lectures on discrete geometry, volume 212 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. Google ScholarDigital Library
- Laurent Miclo. On eigenfunctions of Markov processes on trees. Probability Theory and Related Fields, 142(3--4):561--594, 2008.Google Scholar
- Andrew Ng, Michael Jordan, and Yair Weiss. On spectral clustering: Analysis and an algorithm. In NIPS'02, 2002.Google Scholar
- Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In STOC, pages 755--764, New York, NY, USA, 2010. ACM. Google ScholarDigital Library
- Prasad Raghavendra, David Steurer, and Prasad Tetali. Approximations for the isoperimetric and spectral profile of graphs and related parameters. In STOC, pages 631--640, New York, NY, USA, 2010. ACM. Google ScholarDigital Library
- Alistair J. Sinclair and Mark R. Jerrum. Approximative counting, uniform generation and rapidly mixing Markov chains. Information and Computation, 82(1):93--133, 1989. Google ScholarDigital Library
- Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell., 22(8):888--905, 2000. Google ScholarDigital Library
- David A. Tolliver and Gary L. Miller. Graph partitioning by spectral rounding: Applications in image segmentation and clustering. In CVPR '06: Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 1053--1060. IEEE Computer Society, 2006. Google ScholarDigital Library
Index Terms
- Multi-way spectral partitioning and higher-order cheeger inequalities
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